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Harper's  Stereotype  Edition. 


THE 


THEORY    AND    PRACTICE 


SURVEYING; 


CONTAINING 

ALL  THE  INSTRUCTIONS  REQUISITE  FOR  THE  SKILFUL 
PRACTICE  OF  THIS  ART. 


WITH   A   NEW    SET    OF  ACCURATE 

MATHEMATICAL  TABLES. 


BY   ROBERT    GIB  SON. 


ILLUSTRATED   BY   COPPER-PLATES. 


NEWLY  ARRANGED,  IMPROVED,  AND  ENLARGED,  WITH  USEFUL 
SELECTIONS, 

BY  JAMES  RYAN, 

AUTHOR   Of   AN   ELEMENTARY    TREATISE    ON   ALGEBRA;    THE    NEW   AMERICAN 

GRAMMAR   OF    ASTRONOMY  ;    T^E    DIFFERENTIAL   AND 

INTEGRAL    CALCULUS,    &C.  &.C. 


/I , 


NEW-YORK : 

PRINTED  AND  PUBLISHED  BY  J.  &  J.  HARPER, 

NO.  82   CLIFF-STREET. 

ANTJ    SOLD   BY    THE    PRINCIPAL    BOOKSELLERS    THROUGHOUT    THE 

UNITED    STATES. 

1833. 


[Entered,  according  to  the  Act  of  Congress,  in  the  year  one  thousand  eight 
hundred  and  thirty ^Iwo,  by  J.  &  J.  Harper,  in  the  Clerk's  Office  of  the  District 
Court  of  the  United  States  for  the  Southern  District  of  New-York.] 


P  R  E  F  A  0-B. 


THE  word  Geometry  imports  no  more  than  to  measure 
the  earth,  or  to  measure  the  land;  yet,  in  a  larger  and 
more  proper  sense,  it  is  applied  to  all  sorts  of  dimensions. 
It  is  generally  supposed  to  have  had  its  rise  among  the 
Egyptians,  from  the  river  Nile's  destroying  and  confound- 
ing all  their  landmarks  by  its  annual  inundations,  which  laid 
them  under  the  necessity  of  inventing  certain  methods  and 
measures  to  enable  them  to  distinguish  and  adjust  the  limits 
of  their  respective  grounds  when  the  waters  were  with- 
drawn. And  this  opinion  is  not  entirely  to  be  rejected, 
when  we  consider  that  Moses  is  said  to  have  acquired 
this  art  when  he  resided  at  the  Egyptian  court.  And 
Achilles  Tatius,  in  the  beginning  of  his  introduction  to 
Aratus's  Phenomena,  informs  us  that  the  Egyptians  were 
the  first  who  measured  the  heavens  and  the  earth,  and 
of  course  the  earth  first ;  and  that  their  science  in  this 
matter  was  engraven  on  columns,  and  by  that  means  de- 
livered to  posterity. 

It  is  a  matter  of  some  wonder,  that  though  Surveying 
appears  to  have  been  the  first,  or  at  least  one  of  the  first, 
of  the  mathematical  sciences,  the  rest  have  met  with 
much  greater  improvements  from  the  pens  of  the  most 
eminent  mathematicians,  while  this  seems  to  have  been 
neglected ;  insomuch  that  I  have  not  been  able  to  meet 
with  one  author  who  has  sufficiently  explained  the  whole 
art  in  its  theory  and  practice.  For  the  most  part,  it  has 
been  treateo!  of  in  a  practical  manner  only ;  and  the  few 
who  have  undertaken  the  theory  have  in  a  great  mea- 
sure omitted  the  practice. 

These  considerations  induced  me  to  attempt  a  method- 
ical, easy,  and  clear  course  of  Surveying :  how  far  I  have 
succeeded  in  it  must  be  determined  by  the  impartial 
reader.  The  steps  I  have  taken  to  render  the  whole 
evident  and  familiar  are  as  follow : — 


202373, 


VI  PREFACE. 


section  Ike  first  (Part  the  First)  ^uuliafL  Decimal 
Fractions.     <fV>  second  section   sgntninar  Involution  and 


Evolution.  Tfee  third  section  4;«mtainn-  the  nature  and 
power  of  Logarithms,  witrT  their  application,  and  the 
method  of  computing  them. 

'The  fourth  section  ^oateinB-  geometrical  definitions, 
theorems,  and  problems,  with  the  description  and  use 
of  the  sector,  Gunter's  scale,  and  other  mathematical 
drawing  instruments  used  by  surveyors. 

Jhe  fifth  section^VontainEK Plane  Trigonometry,  right- 
angled  and  oblique,  with  a  variety  of  rules  and  practical 
examples. 

The  first  section  (Part  the  Second)  gives  an  account 
of  the  chains  and  measures  used  in  Great  Britain  and 
Ireland,  methods  of  surveying  and  of  taking  inaccessible 
distances  by  the  chain  only,  with  some  necessary  prob- 
lems ;  also  a  particular  description  of  the  several  instru- 
ments used  in  surveying,  with  their  respective  uses. 

The  second  section  contains  the  mensuration  of  heights 
and  distances,  with  a  great  variety  of  problems  and  prac- 
tical examples. 

The  third  section  contains  the  mensurajtion  of  areas, 
or  the  various  methods  of  calculating  the  superficial  con- 
tents of  any  field  ;  also  several  new  rules  and^problems, 
with  practical  examples,  and  various  methods  of  finding 
the  areas  of  maps  from  their  geometrical  construction  ; 
two  of  which,  more  concise  than, the  rest,  were  first  pub- 
lished in  this  work.  Also,  it  contains  four  new  and  much 
more  concise  methods  of  determining  the  areas  of  sur- 
veys from  the  field-notes,  or  by  calculation,  than  any 
hitherto  published ;  to  these  is  added  the  method  of  cal- 
culating the  area  of  a  survey,  by  having  the  meridian 
pass  through  the  east  or  west  point  of  the  survey,  with 
the  method  of  discovering  these  points  from  the  field- 
notes,  and  the  method  of  correcting  the  errors  by  the 
pen,  when  the  survey  does  not  close :  also  another  new 
method  for  calculating  the  area,  by  having  a  parallel  of 
latitude  pass  through  the  north  or  south  point  of  the 
survey.  The  whole  geometrically  considered  and  de- 
monstrated.* 

*  The  remaining  part  of  the  Author's  Preface  I  have  altered  according 
to  the  arrangement  and  improvement  of  this  new  edition. — EDITOR. 


PREFACE.  Vil 

The  fourth  section  contains  the  nature  of  offsets,  and 
the  method  of  casting  them  up  by  the  pen. 

The  fifth  section  contains  the  method  of  finding  the 
areas  by  intersections. 

The  sixth  section  shows  how  to  enlarge  or  diminish  a 
map,  or  to  reduce  a  map  from  one  scale  to  another ;  also 
the  manner  of  uniting  separate  maps  of  lands  which  join 
each  other  into  one  map  of  any  assigned  size. 

The  seventh  section  contains  the  method  of  dividing 
land,  or  of  taking  off  or  enclosing  any  given  quantity. 

Section  the  eighth  treats  of  surveying  harbours,  shoals, 
sands,  &c. 

Section  the  ninth  treats  of  levelling,  adapted  to  the 
surveying  of  roads  and  hilly  ground,  with  promiscuous 
questions. 

Section  the  first  (Part  the  Third)  contains  the  astro- 
nomical methods  of  finding  the  latitude,  variation  of  the 
compass,  &c.,  with  a  description  of  the  instruments  used 
in  these  operations. 

Section  the  second  contains  a  description  of  the  instru- 
ments requisite  in  astronomical  observations. 

Sectiqn  the  third  shows  how  to  find  the  variation, of 
the  compass ;  with  a  description  of  the  azimuth  compass, 
and  its  use. 

In  this  edition  is  introduced  a  new  set  of  accurate 
Mathematical  Tables. 

Truth  calls  upon  me  to  acknowledge,  that  the  methods 
of  calculation  herein  set  forth  got  their  rise  from  those 
of  the  late  Thomas  Burgh,  Esq.*  who  first  discovered  a 
universal  method  for  determining  the  areas  of  right-lined 
figures,  and  for  which  he  obtained  a  reward  of  twenty 
thousand  pounds  sterling  from  the  Irish  Parliament.  I" 
hope,  therefore,  it  cannot  be  construed  as  an  intention  in 
me  to  take  from  his  great  merit  when  I  say,  that  the 
methods  herein  contained  are  much  more  concise  and 
ready  than  his. 

*  This  method,  with  very  little  alteration  and  improvement,  in  this 
country,  is  usually  called  the  Pennsylvania  Method  of  Calculation. — ED. 


CONTENTS. 


PART  I. 

Sect.  Page 

1.  Decimal  Fractions  .  .     11 

2.  Involution  and  Evolu- 

tion       22 

3.  Of  Logarithms  ....     28 

4.  Elements  of  Geome- 

try .  .  .' 40 

Mathematical  Instru- 
ments      64 

5.  Trigonometry    ....  82 

PART  H. 

1.  The  Chain 109 

The  Circumferentor  .   121 
The  Theodolite  ...  125 
The  Semicircle    ...  128 
The  Plane  Table  .  .     ib. 
Mensuration    of   An- 
gles by  these  Instru- 
ments     131 

The  Protractor    .  .  .     ib. 

2.  Mensuration  of  Heights  137 
Of  Distances    .  146 

3.  Mensuration  of  Areas  151 
General  Method  ...   177 
Pennsylvania  Method   187 
Of  computing  the  A- 

rea  of  a  Survey,  ge- 
ometrically  corisid- 

A3 


Sect  .   ^         Page 

ered     and    demon- 
strated .......  191 

4.  Of  Offsets 200 

5.  Method  of  Surveying 

by  Intersections   .  .  205 

6.  Changing  the  Scale  of 

Maps 208 

7.  Method    of    dividing 

Land 213 

8.  Maritime  Surveying  .  220 

9.  Levelling 222 

Promiscuous      Ques- 

•  tions 229 

PART  IE. 

It  Introductory     Princi- 
ples     231 

2.  Description  of  Instru- 

ments     235 

3.  Variation  of  the  Com- 

'  pass 243 

LIST  OF  TABLES, 

Logarithms  of  Numbers. 
Sines  and  Tangents. 
Traverse  Table. 
Natural  Sines. 


EXPLANATION 

OF  THE  MATHEMATICAL  CHARACTERS  USED  IN  THIS  WORK 


+  signifies  plus,  or  addition. 

"     minus,  or-  subtraction. 
X   or  .  "     multiplication, 
-r  "     division. 

:  : :  :     "     proportion. 
=  "     equality. 

V  "     square  root. 

f/  "     cube  root,  &c. 

05          "     difference  between  two  numbers,  when  it  is  not 
known  which  is  the  greater. 


Thus, 


5  +  3,  denotes  that  3  is  to  be  added  to  5. 

6  —  2,  denotes  that  2  is  to  be  taken  from  6. 

7  X  3,  or  7  .  3,  denotes  that  7  is  to  be  multiplied  by  3. 

8  -r-  4,  denotes  that  8  is  to  be  divided  by  4. 
2:3  :  :  4:6,  shows  that  2  is  to  3  as  4  is  to  6. 

6  +  4  =  10,  shows  that  the  sum  of  6  and  4  is  equal  to  10. 

\/  3,  or  3  ,  denotes  the  square  root  of  the  number  3. 

y'S,  or  53,  denotes  the  cube  root  of  the  number  5. 
72,  denotes  that  the  number  7  is  to  be  squared. 
83,  denotes  that  the  number  8  is  to  be  cubed. 
Et  cetera* 


OF  THE 

UNIVERSITY 

OF 


THE 

THEORY    AND    PRACTICE 

OF 

SURVEYING. 


THE  word  Surveying,  in  the  mathematics,  signifies  the  art 
of  measuring  land,  and  of  delineating  its  boundaries  on  a  map. 

The  Surveyor,  in  the, practice  of  this  art,  directs  his  attention, 
at  first,  to  the  tracing  and  measuring  of  lines ;  secondly,  to 
the  position  of  these  lines  in  respect  to  each  other,  or  the  angles 
formed  by  them ;  thirdly,  to  the  plan,  or  representation  of  the 
field  or  tract  which  he  surveys  ;  and  fourthly,  to  the  calculation 
of  its  area,  or  superficial  content.  When  this  art  is  employed 
in  determining  the  variation  of  the  compass,  in  observing  and 
delineating  coasts  and  harbours,  their  latitude,  longitude,  and 
soundings,  together  with  the  bearings  of  their  most  remark- 
able places  from  each  other,  it  is  usually  denominated  Maritime 
Surveying.  This  branch  of  Surveying,  however,  demands  no 
other  qualifications  than  those  which  should  be  thoroughly 
acquired  by  every  land-surveyor  who  aspires  to  the  character 
of  an  accomplished  and  skilful  practitioner.  Surveying,  there- 
fore, requires  an  intimate  acquaintance  with  the  several  parts 
of  the  mathematics  which  are  here  inserted  as  an  introduction 
to  this  treatise. 


PART  I. 

Containing  Decimal  Fractions,  Involution  and  Evolution,  the 
Nature  and  Use  of  Logarithms,  Geometry,  and  Plane  Trigo- 
nometry. 

SECTION  I. 

DECIMAL   FRACTIONS. 

If  we  suppose  unity  or  any  one  thing  to  be  divided  into  any 
assigned  number  of  equal  parts,  this  number  is  called  the  de- 


12  DECIMAL  FRACTIONS. 

nominator ;  and  if  we  choose  to  take  any  number  of  such  parts 
less  than  the  whole,  this  is  called  the  numerator  of  a  fraction. 

The  numerator,  in  the  vulgar  form,  is  always  written  over 
the  denominator,  and  these  are  separated  by  a  small  line  thus 
f ,  or  £  ;-  the  first  of  these  is  called  three-fourths,  and  the  latter 
five-eighths,  of  an  inch,  yard,  &c.,  or  of  whatever  the  whole 
thing  originally  consisted :  the  4  and  the  8  are  the  denominators, 
showing  into  how  many  equal  parts  the  unit  is  divided  ;  and  the 
three  and  the  live  are  the  numerators,  showing  how  many  of 
those  parts  are  under  consideration. 

Fractions  are  expressed  in  two  forms,  that  is,  either  vulgarly 
or  decimally. 

AH  fractions  whose  denominators  do  not  consist  of  a  cipher 
or  ciphers,  set  after  unity,  are  called  vulgar  ;  and  their  denomi- 
nators are  always  written  under  their  numerators.  The  treat- 
ment of  these,  however,  would  be  foreign  to  our  present  purpose. 
But  fractions  whose  denominators  consist  of  a  unit  prefixed 
to  one  or  more  ciphers,  are  called  decimal  fractions  ;  the  nume- 
rators of  which  are  written  without  their  denominators,  and  are 
distinguished  from  integers  by  a  point  prefixed ;  thus  T\,  T4^, 
Tyj_,  in  the  decimal  form,  are  expressed  by  .2,  .42,  .172. 

The  denominators  of  such  fractions  consisting  always  of  a 
unit  prefixed  to  as  many  ciphers  as  there  are  places  of  figures 
in  the  numerators,  it  follows,  that  any  number  of  ciphers  put 
after  those  numerators,  will  neither  increase  nor  lessen  their 
valjie :  for  T3¥,  T\07,  and  TYo-V  are  all  °f  tne  same  value,  and 
will  stand  in  the  decimal  form  thus  .3,  .30,  .300 ;  but  a  cipher 
or  ciphers  prefixed  to  those  numerators  lessen  their  value  in  a 
tenfold  proportion :  for  r\,  T°/¥,  and  yVro*  which  in  the  decimal 
form  we  denote  by  .3,  .03,  and  .003,  are  fractions,  of  which 
the  first  is  ten  times  greater  than  the  second ;  and  the  second, 
ten  times  greater  than  the  third. 

Hence  it  appears,  that  as  the  value  and  denomination  of  any 
figure,  or  number  of  figures,  in  common  arithmetic  is  enlarged 
and  becomes  ten,  or  a  hundred,  or  a  thousand  times  -greater, 
by  placing  one,  or  two,  or  three  ciphers  after  it ;  so  in  decimal 
arithmetic,  the  value  of  any  figure,  or  number  of  figures,  de- 
creases and  becomes  ten,  or  a  hundred,  or  a  thousand  times 
less,  while  the  denomination  of  it  increases,  and  becomes  so 
many  times  greater,  by  prefixing  one,  or  two,  or  three  ciphers 
to  it :  and  that  any  number  of  ciphers  before  an  integer,  or 
after  a  decimal  fraction,  has  no  effect  in  changing  its  value. 


DECIMAL  FRACTIONS.  13 

SCALE    OF    NOTATION. 

Integers.  Decimals. 


7342186     875326 

ifffff!    f  inn 

H;if**      Hffrlf 

girli.  -88.188.1 

*       GO      C.  &•  CO  >-t     £.'*  ~?  <*>     e+    ~ 

t\  * '  011 

g    a*  ™*S    gf 

to   *  3-  cr 

I  '"S 

ADDITION  OF  DECIMALS. 

"Write  the  numbers  under  each  other  according  to  the  value 
or  denomination  of  their  places ;  which  position  will  bring  all 
the  decimal  points  into  a  column,  or  vertical  line,  by  themselves. 
Then,  beginning  at  the  right-hand  column  of  figures,  add  in 
the  same  manner  as  in  whole  numbers,  and  put  the  decimal 
point  in  the  sum  directly  beneath  the  other  points. 

EXAMPLES. 

Add  4.7832,  3.2543, 7.8251, 6.03, 2.857,  and  3.251  together. 
Place  them  thus, 

4.7832 

3.2543 

7.8251 

6.03 

2.857 

3.251 

Sum  =  28.0006 

Add  6.2,  121.306,  .75, 2.7,  and  .0007  together. 
121.306 
.75 

2.7 
.0007 


Sum=  130.9567 


What  is  the  sum  of  6.57,  1.026,  .75, 146.5,  8.7,  526.,  3.97, 
and  .0271? 

Answer,  693.5431, 


14  DECIMAL  FRACTIONS. 

What  is  the  sum  of  4.51, 146.071,  .507,  .0006, 132.,  62.71, 
.507,  7.9,  and  .10712? 
Answer,  354.31272. 

SUBTRACTION  OF  DECIMALS. 

Write  the  figures  of  the  subtrahend  beneath  those  of  the 
minuend  according  to  the  denomination  of  their  places,  as  di- 
rected in  the  rule  of  addition  ;  then,  beginning  at  the  right-hand, 
subtract  as  in  whole  numbers,  and  place  the  decimal  point  in 
the  difference  exactly  under  the  other  two  points. 

EXAMPLES. 

From  38.765  take  25.3741 
25.3741 

Difference  =  13.3909 


From  2.4  take  .8473 
.8472 

Diff.  =  1.5528 


From  71.45  take  8.4837248. 
Difference  =  62.9662752. 
From  84  take  82.3412. 
Diff.=  1.6588. 

MULTIPLICATION  OF  DECIMALS. 

Set  the  multiplier  under  the  multiplicand  without  any  regard 
to  the  situation  of  the  decimal  point ;  and  having  multiplied  as 
in  whole  numbers,  cut  off  as  many  places  for  decimals  in  the 
product,  counting  from  the  right-hand  towards  the  left,  as  there 
are  in  both  the  multiplicand  and  multiplier :  but  if  there  be  not  a 
sufficient  number  of  places  in  the  product,  the  defect  may  be 
supplied  by  prefixing  ciphers  thereto. 

For  the  denominator  of  the  product  being  a  unit,  prefixed 
to  as  many  ciphers  as  the  denominators  of  the  multiplier  and 
multiplicand  contain  of  ciphers,  it  follows  that  the  places  of  de- 
cimals in  the  product  will  be  as  many  as  in  the  numbers  from" 
whence  it  arose. 


DECIMAL  FRACTIONS.  15 

EXAMPLES.      x 

Multiply  4«.765  by  .003609. 
.003609 

438885 
292590 
146295 


Product  =  .175992885 


Multiply  .121 
by    .14 


Product  =  .01694 

Multiply  121.6  by  2.76 
2.76 

7296 
8512 
2432 

Product  =  335.616 

Multiply  .0089789  by  1085. 

Product  =  9.7421065. 
Multiply  .248723  by  .13587. 

Product  =  .03379399401. 

DIVISION  OF   DECIMALS. 

Divide  as  in  whole  numbers ;  observing  that  the  divisor  and 
quotient  together  must  contain  as  many  decimal  places  as  there 
are  in  the  dividend.  If,  therefore,  the  dividend  have  just  as 
many  places  of  decimals  as  the  divisor  has,  the  quotient  will 
be  a  whole  number  withouj.  any  decimal  figures.  If  there  be 
more  places  of  decimals  in  the  dividend  than  there  are  in  the 
divisor,  point  off  as  many  figures  in  the  quotient  for  decimals, 
as  the  decimal  places  in  the  dividend  exceed  those  in  the  divisor ; 
the  want  of  places  in  the  quotient  being  supplied  by  prefixing 
ciphers.  But  if  there  be  more  decimalplaces  in  the  divisor  than 
in  the  dividend,  annex  ciphers  to  the  dividend,  so  that  the  decimal 
places  here  may  be  equal  in  number  to  those  in  the  divisor; 
and  men  the  quotient  will  be  a  whole  number,  without  fractious* 


16  DECIMAL  FRACTIONS. 

When  there  is  a  remainder,  after  the  division  has  been  thus 
performed,  annex  ciphers  to  this  remainder,  and  continue  the 
operation  till  nothing  remains,  or  till  a  sufficient  number  of 
decimals  shall  be  found  in  the  quotient. 

EXAMPLES. 

Divide  .144  by  .12. 

.12).144(1.2  =  quotient. 
12 
' 

24 
24 

*~--.^.  ^ 

0 

Divide  63.72413456922  by  2718.^ 

2718)63.72413456922(.02344522979  =  quotient. 
5436 

9364 
8J54 

12101 
10872 

12293 
1087$ 

14214 
13590 

6245 
5436 

8096 
5436 

26609 
24462 

21472 
19026 

24462 
24462 


DECIMAL  FRACTIONS.  17 

There  being  11  decimal  figures  in  the  dividend,  and  none  in 
the  divisor,  1 1  figures  are  to  be  cut  off  in  the  quotient ;  but  as 
the  quotient  itself  consists  of  but  10  figures,  prefix  to  them  a 
cipher  to  complete  that  number. 

Divide  1.728  by  .012 

.012)1.728(144  =  quotient.— 
12 

52 
48 

48 
48 

——  £*. 

~ 

Because  the  number  of  decimal  figures  in  the  divisor  and 
dividend  are  alike,  the  quotient  will  be  integers. 

Divide  2  by  3.1416 

3.1416)2.0000,0(0.636618+  =  quotient. 
1  8849  6 


115040 
94248 

207920 
188496 

194240 

188496 

57440 
31416 

260240 
251328 

8912+ 

In  this  example  there  are  four  decimal  figures  in  the  divisor, 
and  none  in  the  dividend  ;  therefore,  according  to  the  rule,  four 
ciphers  are  annexed  to  the  dividend,  which,  in  this  condition,  is 
yet  less  than  the  divisor.  A  cipher  must  then  be  put  in  the 
quotient  in  the  place  of  integers,  and  other  ciphers  annexed  to 
the  dividend ;  and  the  division  being  now  performed,  the  deci- 
mal figures  of  the  quotient  are  obtained. 


;*,,. 

18  DECIMAL  FRACTIONS. 

Divide  7234.5  by  6.5        Quotient  =  1113. 

Divide  476.520  by  .423  =  1126.5+ 

Divide  .45695  by  12.5     • =  .0365+ 

Divide  2.3  by  96  =  .02395+ 

Divide  87446071  by  .004387    —  =  19933000000 
Divide  .624672  by  482    =  .001296. 

REDUCTION  OF  DECIMALS 

RULE    I. 

To  reduce  a  Vulgar  Fraction  to  a  Decimal  of  the  same  value. 
Having  annexed  a  sufficient  number  of  ciphers,  as  decimals, 
to  the  numerator  of  the  vulgar  fractions,  divide  by  the  denomi- 
nator ;  and  the  quotient  thence  arisjng  will  be  the  decimal  frac 
tion  required. 

EXAMPLE. 

Reduce  f  to  a  decimal  fraction. 
4)3.00 


.75=decirnal  required. 

For  f  of  one  acre,  mile,  yard,  or  any  thing,  is  equal  to  1  of 
3  acres,  miles,  yards,  &c. ;  therefore  if  3  be  divided  by  4,  the 
quotient  is  the  answer  required. 
Reduce  f  to  a  decimal  fraction.  Answer  .4 

Reduce  if .48 

Reduce  ^  -  ....      .1146789 

Reduce  % .7777+ 

Reduce  fi  -      .9130434+ 

Reduce  |,  i,  {,  |,  and  so  on  to  Jg-,  to  their  corresponding 
decimal  fractions ;  and  in  this  operation  the  various  modes  of 
interminate  decimals  may  be  easily  observed. 

RULE    II. 

To  reduce  Quantities  of  the  same,  or  of  different  Denominations^ 

to  Decimal  Fractions  of  higher  Denominations. 
If  the  given  quantity  consist  of  one  denomination  only,  write 
it  as  the  numerator  of  a  vulgar  fraction ;  then  consider  how 
many  of  this  make  one  of  the  higher  denomination,  men- 
tioned in  the  question,  and  write  this  latter  number  under  the 
former,  as  the  denominator  of  a  vulgar  fraction.  When  this 
has  been  done,  divide  the  numerator  by  the  denominator,  as 
directed  in  the  foregoing  rule,  and  the  quotient  resulting  will  be 
the  decimal  fraction  required. 


DECIMAL  FRACTIONS.  19 

But  if  the  given  quantity  contain  several  denominations,  re- 
duce them  to  the  lowest  term  for  the  numerator ;  reduce  likewise 
that  quantity  whose  fraction  is  sought  to  the  same  denomina- 
tion, for  the  denominator  of  a  vulgar  fraction ;  then  divide  as 
before  directed. 

EXAMPLES. 

Reduce  9  inches  to  the  decimal  of  a  foot. 
The  foot  being  equal  to  12  inches,  the  vulgar  fraction  will 
be-^;  then  12)9:00 

.75=decimal  fraction  required. 
Reduce  8  inches  to  the  decimal  of  a  yard. 
8  inches 


1  yard  X  3  X  12  =  36 

36)8.0(.22+  =  Answer. 
72 

80 
72 

8 

Reduce  5  furlongs  00  perches  to  the  decimal  of  a  mile. 
1  mile  5  furlongs 

8  40 

200 

=  vulgar  fraction. 

320 

320  per. 

320)200.0(.625  =  decimal  sought. 
1920 

800 
640 

1600 
1600 

Reduce  21  minutes  54  seconds  to  the  decimal  of  a  degree. 
Ans.  .365. 

Reduce  .056  of  a  polejx>  the  decimal  of  an  acre.  Ans.  .00035. 

Reduce  13  cents  to  the  decimal  of  an  eagle.     Ans.  .013. 

Reduce  1 4  minutes  to  the  decimal  of  a  day.     Ans.  .00972+ 

Reduce  3  hours  46  minutes  to  the  decimal  of  a  week.  Ans* 
0224206+ 


20  DECIMAL  FRACTIONS. 


RULE  in. 

To  find  the  value  of  Decimal  Fractions  in  terms  of  the  lower 
denominations. 

Multiply  the  given  decimal  by  the  number  of  the  next  lower 
denomination  which  makes  an  integer  of  the  present,  and  point 
off  as  many  places  at  the  right-hand  of  the  product,  for  a  re- 
mainder, as  there  are  figures  in  the  given  decimal.  Multiply 
this  remainder  by  the  number  of  the  next  inferior  denomination, 
and  point  off  a  remainder  as  before.  Proceed  in  this  manner 
through  all  the  parts  of  the  integer,  and  the  several  denomina- 
tions standing  on  the  left-hand  are  the  value  required. 

EXAMPLES. 

Required  the  value  of  .3375  of  an  acre. 

4  =  number  of  roods  in  an  acre. 

1.3500 

40  =  number  of  perches  in  a  rood. 

14.0000 

The  value,  therefore,  is  1  rood  14  perches. 
What  is  the  value  of  .6875  of  a  yard  ? 

3  =  number  of  feet  in  a  yard. 

2.0625 

12  =  number  of  inches  in  a  foot. 

.7500 

12  =  number  of  lines  in  an  inch. 


9.0000 

The  answer  here  is  2  feet  9  lines. 

What  is  the  value  of  .084  of  a  furlong?  Ans.  3  per.  1  yd. 
2  ft.  11  in. 

What  is  the  value  of  .683  of  a  degree  ?  Ans.  40  m.  58  sec. 
48  thirds. 

What  is  the  value  of  .0053  of  a  mile  ?  Ans.  1  per.  3  yds. 
2  ft.  5  in.+ 

What  is  the  value  of  .036  of  a  day  ?    Ans.  61'  50"  24'" 


DECIMAL  FRACTIONS.  21 


PROPORTION  IN  DECIMAL  FRACTIONS. 

Having  reduced  all  the  fractional  parts  in  the  given  quantities 
to  their  corresponding  decimals,  and  having  stated  the  three 
known  terms,  so  that  the  fourth,  or  required  quantity,  may  be  as 
much  greater  or  less  than  the  third  as  the  second  term  is 
greater  or  less  than  the  first,  then  multiply  the  second  and 
third  terms  together,  and  divide  the  product  by  the  first  term, 
and  the  quotient  will  be  the  answer ; — in  the  same  denomination 
with  the  third  term. 

EXAMPLES. 

If  3  acres  3  roods  of  lanoT  can  be  purchased  for  93  dollars 
60  cents,  how  much  will  15  acres  1  rood  cost  at  that  rate  ? 

3  acs.  3  rds.  =  3.75  acres. 
15  acs.  1  rd.  =  15.25  acres. 
$93,  60  cents  =$93.60 

Then  3.75  :  15.25  :  :  93.60: 
15.25 

46800 
18720 
46800 

9360 

^  g 

3.75)1427.4000(380.64  =  Answer. 
1125 

3024 
3000 

2400 
2250 

1500 

1500 

I 

If  a  clock  gain  14  seconds  hi  5  days  6  hours,  how  much 
.  will  it  gain  in  17  days  15  hours?     Ans.  47  seconds. 

If  187  dollars  85  cents  gain  12  dollars  33  cents  interest  in 
a  year,  at  what  rate  per  cent,  is  this  interest?  Ans.  6.56+ 


22  INVOLUTION  AND  EVOLUTION. 

SECTION  II. 

INVOLUTION  AND  EVOLUTION. 

INVOLUTION  is  the  method  of  raising  any  number,  considered 
as  the  root,  to  any  required  power. 

Any  number,  whether  given  or  assumed  at  pleasure,  may 
be  called  the  root  or  first  power  of  this  number ;  and  its  other 
powers  are  the  products  that  result  from  multiplying  the  number 
by  itself,  and  the  last  product  by  the  same  number  again,  and 
so  on  to  any  number  of  multiplications. 

The  index,  or  exponent,  is  the  number  denoting  the  height, 
or  degree  of  the  power,  being  always  greater  by  one  than  the 
number  of  multiplications  employed  in  producing  the  power. 
It  is  usually  written  above  the  root,  as  in  the  following  EX- 
AMPLE, where  the  method  of  involution  is  plainly  exhibited. 

Required  the  fifth  power  of  8  =  the  root,  or  first  power, 
first  multiply  by  -     -     8 

then  multiply  the  product  64  =  82  =  square,  or  second  power, 
by         8 

&c.  512  =  83  =  cube,  or  third  power. 
8 

4096  =  84  =  biquadrate,  or  fourth  power. 
8 


32768  =  85  =  Answer. 

EXAMPLES    FOR  .EXERCISE. 

What  is  the  second  power  of  3.05  ?     Ans.  9.3025. 
What  is  the  third  power  of  85.3  ?     Ans.  620650.477. 
What  is  the  fourth  power  of  .073  ?     Ans.  .000028398241. 
What  is  the  eighth  power  of  .09  ?    Ans.  .00.00.00.0043046721. 

Note. — When  two  or  more  powers  are  multiplied  together, 
their  product  is  that  powet  whose  index  is  the  sum  of  the  in- 
dices of  the  factors,  or  powers  multiplied. 

„    EVOLUTION  is  the  method  of  extracting  any  required  root 
from  any  given  power. 

Any  number  may  be  considered  as  a  power  of  some  other 
number;  and  the  required  root  of  any  given  power  is  that 


EVOLUTION.  23 

number  which  being  multiplied  into  itself  a  particular  number 
of  times  produces  the  given  power;  thus  if  81  be  the  given 
number,  or  power,  its  square  or  second  root  is  9 ;  because  9  X 
9=9*  =81',  and  3  is  its  biquadrate,  or  fourth  root,  because 
3X3X3X3=34=81.  Again,  ifj!729  be  the  given  power,  and 
its  cube  root  be  required,  the  answer  is  9,  for  9  X  9  X  9=729  ; 
and  if  the  sixth  root  of  that  number  be  required,  it  is  found  to 
be  3,  for  3  x  3  x  3  X  3  X  3  X  3=729. 

The  required  power  of  any  given  number,  or  root,  can 
always  be  obtained  exactly,  by  multiplying  the  number  continu- 
ally into  itself;  but  there  are  many  numbers  from  which  a 
proposed  root  can  never  be  completely  extracted ; — yet  by  ap- 
proximating with  decimals,  these  roots  may  be  found  as  exact 
as  necessity  requires.  The  roots  that  are  found  complete  are 
denominated  rational  roots,  and  those  which  cannot  be  found 
completed,  or  which  only  approximate,  are  called  surd,  or 
irrational  roots. 

Roots  are  usually  represented  by  these  characters  or  ex- 
ponents : 

N/,  or  *  which  signifies  the  square  root;  thus,  \/9,  or  92=3. 

\r  or  ¥  cube  root ;  ^64,  or  64^=4. 

\V  or  *   biquadrate  root;     v'lG,  or  16T=2,  &c. 

Likewise  8¥  signifies  the  square  root  of  8  cubed ;  and,  in 
general,  the  fractional  indices  imply  that  the  given  numbers  are 
to  be  raised  to  such  powers  «s  are  denoted  by  their  numerators, 
and  that  such  roots  are  to  be  extracted  from  these  powers  as 
are  denoted  by  their  denominators. 

RULE 

For  extracting  the  Square  Root. 

Commencing  at  the  unit  figure,  cut  off  pefi*s  of  two  figures 
each,  till  all  the  figures  of  the  given  number  are  exhausted.* 

The  first  figure  of  the  required  root  will  be  the  square  root 

4fr* 

*  In  dividing  a  decimal,  or  a  number  consisting  of  a  whole  number 
with  a  decimal,  into  periods,  the  division  must  also  commence  at  the  unit 
figure  or  decimal  point,  and  must  be  continued  both  ways,  if  there  be  a 
whole  number ;  and  if  there  be  an  odd  figure  at  the  end  of  the  deciijMil,  a 
cipher,  or  if  it  be  a  periodical  decimal,  the  figure  that  would  next  atise, 
from  its  continuation,  must  be  annexed  ;  thus  417.245  will  be  divided 
thus,  4'17'.24/50:  41.66666,  &c.  thus,  41/.66/66'66  :  and  .567  thus, 
56'70,  &c. 

See  the  Editor's  "  Elementary  Treatise  on  Arithmetic,  in  Theory  and 
Prac/zce,".page  219. — ED. 


24        .  EVOLUTION.      ^ 

of  the  first  period,  or  of  the  greatest  square  root  contained  in  , 
it,  if  it  be  not  a  square  itself. 

Subtract  the  square  of  this  figure  from  the  first  period ;  to 
the  remainder  annex  the  next  period  for  a  dividend ;  and  for 
part  of  a  divisor,  double  tjbie  part  of  the  root  already  obtained. 

Try  how  often  this  part  of  the  divisor  is  contained  in  the 
dividend  wanting  the  last  figure,  and  annex  the  figures  thus 
found  to  the  parts  of  the  root  and  of  the  divisor  already  de- 
termined. 

Thus  multiply  and  subtract  as  in  division ;  to  the  remainder 
bring  down  the  next  period,  and,  adding  to  the  divisor  the  figure 
of  the  root  last  found,  proceed  as  before.* 

If  any  thing  remain  after  continuing  the  process  till  all  the 
figures  in  the  given  number  have  been  used,  proceed  in  the  same 
manner  to  find  decimals,  adding,  to  find  each  figure,  two  ciphers, 
pr  if  the  given  number  end  in  an  interminate  decimal,  the  two 
figures  that  would  next  arise  from  its  continuation. 

To  extract  the  root  of  a  fraction,  reduce  it  to  its  simplest  form, 
if  it  be  not  so  already,  and  extract  the  root  of  both  'terms,  if 
they  be  complete  powers :  otherwise  divide  the  root  of  their 
product  by  the  denominator. 

The  root  may  also  be  found  by  reducing  the  fraction  to  a 
decimal,  if  it  be  not  such  already,  and  taking  the  root  of  the 
decimal. 

"%  .   s  ' 

EXAMPLES. 

Required  the  square  root  of  1710864. 

1'71'08'64' 

1710864(1308  =  Answer. 
1 


23 


71 
69 


2608  I  20864 
20864 


*  The  principle  on  which  the  preceding  rule  depends  is,  that  the  square 
of  the  sum  of  two  numbers  is  equal  to  the  squares  of  the  numbers  with  twice 
their  product.  Thus,  the  square  of  34  is  equal  to  the  squares  of  30  and 
of  4  with  twice  the  product  of  30  and  4 ;  that  is,  to  900-J-2  X30  x4-[-16= 
1156.  Here,  in  extracting  the  second  root  of  1156,  we  separate  it  into 
two  parts,  1 100  and  56.  Thus  1 100  contains  900,  the  square  of  30,  with 
the  remainder  200  ;  the  first  part  of  the  root  is  therefore  30,  and  the  re- 
mainder 200-J-56,  or  256.  Now,  according  to  the  principle  above  men- 


EVOLUTION.  25 


Required  the  square  root  of  16007.3104. 

I'60'07'.3r04r 


1 


22 


16007.3104(126.52  =  Answer.' 


60 
44 


246  I  1607 
6  I  1476 

2525  I  13131 
5  I  12625 

25302  I  50604 
50604 


EXAMPLES   FOR   EXERCISE, 

Required  the  square  root  of  298116.     Ans.  546. 
Required  the  square  root  of  348.17320836.     Ans.  18.6594. 
Required  the  square  root  of  17.3056.     Ans.  4.16. 
Required  the  square  root  of  .000729.     Ans.  .027. 
Required  the  square  root  of  17f.     Ans.  4.168333+ 

TO  EXTRACT  THE  CUBE  ROOT. 

RULE  I. — Commencing  at  the  unit  figure,  cut  off  periods  of 
three  figures  each,  till  all  the  figures  of  the  given  number  are 
exhausted.  Then  find  the  greatest  cube  number  contained  in 
the  first  period,  and  place  the  cube  root  of  it  in  the  quotient. 
Subtract  its  cube  from  the  first  period,  and  bring  down  the 
next  three  figures ;  divide  the  number  thus  brought  down  by 
300  times  the  square  of  the  first  figure  of  the  root,  and  it 
will  give  the  second  figure ;  add  300  times  the  square  of  the 
first  figure,  30  times  the  product  of  the  first  and  second  figures, 
and  the  square  of  the  second  figure  together,  for  a  divisor ;  then 

tioned,  this  remainder  must  be  twice  the  product  of  30,  and  the  part  of  the 
root  still  to  be  found,  together  with  the  square  of  that  part.  Now,  dividing 
256  by  00,  the  double  of  30,  we  find  for  quotient  4 ;  then  this  part  being 
added  to  60,  the  sum  is  64,  which  being  multiplied  by  4,  the  product  256  is 
evidently  twice  the  product  of  30  and  4,  together  with  the  square  of  4. 
In  the  same  manner  the  operation  may  be  illustrated  in  every  case.  The 
rule,  however,  is  best  demonstrated  by  Algebra. 

See  my  Treatise  on  this  subject,  page  231,  second  edition. — ED. 
B 


26  EVOLUTION. 

multiply  this  divisor  by  the  second  figure,  and  subtract  the  re- 
sult from  the  dividend,  and  then  bring  down  the  next  period, 
and  so  proceed  till  all  the  periods  are  brought  down.* 

To  extract  the  cube  root  of  a  fraction,  reduce  it  to  a  decimal, 
and  then  extract  the  root ;  or  multiply  the  numerator  by  the 
square  of  the  denominator,  find  the  cube  root  of  the  product,  and 
divide  by  the  denominator. 

The  cube  root  of  a  mixed  number  is  generally  best  found  by 
reducing  the  fractional  part  to  a  decimal,  if  it  be  not  so  already, 
and  then  extracting  the  root.  It  may  be  also  found  by  reducing 
the  given  number  to  an  improper  fraction,  and  then  working 
according  to  the  preceding  directions. 

EXAMPLES. 

1.  Required  the  cube  root  of  48228.544. 


32X  300=2700 
3   X30  =     90 

Divisor  2790 


48'223'.644'(36.41  Root. 


27 


21228  Resolvend. 
19656  Subtrahend. 


32  x  300  X  6  =  1 6200  ) 

3   X    30X62=3240V    1572.544  Resolvend. 
6 3  =  2 1 6  j     1 572. 544  Subtrahend. 


Subtrahend  19656 


362X300=388800 
36    X    30=     1080 


Divisor  389880 
362X  300X4  =1555200) 
36    X    30X42=     17280  \ 
43=  64) 


Subtrahend.  1672544 

Ex.  2.  What  is  the  cube  root  of  62570773  ?  Ans.  397. 

Ex.  3.  What  is  the  cube  root  of  51478848  ?  Ans.  372. 

Ex.  4.  What  is  the  cube  root  of  84.604519?  Ans.  4.39. 

Ex.  5.  What  is  the  cube  root  of  16974593?  Ans.  257. 

*  The  reason  of  this  rule  will  appear  evident  from  the  following  illus- 
tration. The  cube  n'25,  for  instance,  is  equivalent  to  the  cube  of  20  ad- 
ded to  the  cube  of  5,  t  gether  with  the  sum  of  300x4x5+30X2X5x5  , 
or,  wluch  is  the  same  thing,  25  is  equal  to  20+5,  and  therefore  25  cubed 


EVOLUTION.  27 

2.    To  extract  the  Cube  Root  by  another  Method.* 

1.  By   trials  find  the  nearest  rational  cube  to  the  given 
number,  whether  it  be  greater  or  less,  and  call  it  the  assumed  cube. 

2.  Then  say,  by  the  Rule  of  Three,  as  the  sum  of  the  given 
number  and  double  the  assumed  cube  is  to  the  sum  of  the  as- 
sumed, and  double  the  given  number,  so  is  the  root  of  the 
assumed  cube  to  the  root  required,  nearly.     Or,  as  the  first  sum 
is  to  the  difference  of  the  given  and  assumed  cube,  so  is  the 
assumed  root  to  the  difference  of  the  roots,  nearly. 

3.  By  using,  in  like  manner,  the  cube  of  the  root  last  found 
as  a  new  assumed  cube,  another  root  will  be  obtained  still 
nearer.     And  so  on  as  far  as  we  please ;  using  always  the 
cube  of  the  last  found  root  for  the  assumed  cube. 

EXAMPLES. 

1.  To  find  the  cube  root  of  21035.8. 

Here  the  root  is  soon  found  between  27  and  28.  Taking 
therefore  27,  its  cube  is  19683,  which  is  the  assumed  cube. 
Then,  Ig6g3  21035.8 

2  2 

39366          42071.6 
21035.8       19683 

As  60401.8  :    61754.6  :  :  27  :  27.6047, 

is  equal  to  20+5  cubed ;  but  20+5  cubed  is  equivalent  to  8000-J-300X 
4x5+30x2X5x5+125,  or  to  203+(300x4+30x2x5+5x5)x5= 

48228544. 


20X20+5X20 

+5x20+25 

Multiplied,  {  20X20+2X5X20^  =  second  power. 

20X20X20+2X5X20X20+20X25 

+5  X  20  X  20+2  X  20  X  25+125 

20  X  20  X  20+3  X  5  X  20  X  20+3  X  20  X  25+1 25=r3d  power 
or,  SOOO+300  X  4  X  5+30  X  2  X  25+125. 

Here  the  rule  is  evident.  In  the  same  manner,  the  operation  may  be 
illustrated  in  every  case.  For  a  demonstration  of  this  rule  in  general 
terms,  the  reader  is  referred  to  the  Editor's  "  Treatise  on  Algebra,  Theo- 
retical arfd  Practical." — ED. 

*  This  rule  is  found  in  Hutton's  Mathematics.     There  have  been  differ- 
ent rules  given  for  extracting  the  cube  root,  among  which  this,  and  another 
rule  given  in  Pike's  Arithmetic  (by  approximation),  are  very  expeditious. 
B2 


28  OF  LOGARITHMS. 

Therefore  27.6047  is  the  root  nearly. 
Again,  by  repeating  the  operation,  and  taking  27.6047  for 
the  assumed  root,  it  will  give  27.60491  the  root  still  nearer. 

2.  Required  the  cube  root  of  3214?     Ans.  14.75758. 

3.  Required  the  cube  root  of  2  ?     Ans.  1.25992. 

4.  Required  the  cube  root  of  256  ?     Ans.  6.349. 


SECTION  III. 
OF  LOGARITHMS. 

LOGARITHMS  are  a  series  of  numbers,  so  contrived,  that  by 
them  the  work  of  multiplication  may  be  performed  by  addition; 
and  the  operation  of  division  may  be  done  by  subtraction.  Or, 
— Logarithms  are  the  indices,  or  series  of  numbers  in  arith- 
me.tical  progression,  corresponding  to  another  series  of  numbers 
in  geometrical  progression.  Thus, 

0,  1,  2,  3,     4,     5,     6,  &c.  indices  or  logarithms. 

1,  2,  4,  8,  16,  32,  64,  &c.  geometrical  progression. 

Or, 

0,  1,  2,     3,     4,       5,       6,  &c.  ind.  or  log.    „ 

1,  3,  9,  27,  81,  243,  729,  &c.  geometrical  series. 

Or, 

50*,  1,  2,  3,  4,  5,  6,&c.ind.orlog. 

1,  10,  100,  1000,  10000,  100000,  1000000,  &c.  geomet- 
rical series, — where  the  same  indices  serve  equally  for  any 
geometrical  series  or  progression. 

Hence  it  appears  that  there  may  be  as  many  kinds  of  indices, 
or  logarithms,  as  there  can  be  taken  kinds  of  geometrical  series. 
But  the  logarithms  most  convenient  for  common  uses  are  those 
adapted  to  a  geometrical  series  increasing  in  a  tenfold  progres- 
sion, as  in  the  last  of  the  foregoing  examples. 

In  the  geometrical  series  1,  10,  100,  1000,  &c.  if  between 
the  terms  1  and  10  the  numbers  2,  3,  4,  5,  6,  7,  8,  9  were 
interposed,  indices  might  also  be  adapted  to  them  in  an  arith- 

*  In  any  system  of  logarithms  the  log.  of  1  is  0 ;  for  logarithms  may 
be  considered  as  the  exponents  of  the  powers  to  which  a  given  or  inva- 
riable number  must  be  raised,  in  order  to  produce  all  the  common  or 
natural  numbers,  therefore  by  assuming  x°=a,  then  by  squaring  x°=aa 
hence  a2— a,  and  consequently  by  division  a=l,  from  whence  it  is  evi- 
dent tint  the  log.  of  1  is  always  =  0,  in  any  system  ;  for  more  on  this 
subject,  and  the  algebraical  form  of  the  rule  for  computing  logarithms, 
see  Bonnycastle's  Algebra,  page  200,  New- York  edition ;  or  my  Treatise 
on  Algebra,  page  332,  second  edition. — ED. 


OF  LOGARITHMS.  29 

metical  progression,  suited  to  the  terms  interposed  between  1 
and  10,  considered  as  a  geometrical  progression.  Moreover, 
proper  indices  may  be  found  to  all  the  number^,  that  can  be 
interposed  between  any  two  terms  of  the  geometrical  series. 

But  it  is  evident  that  all  the  indices  to  the  numbers  under  10, 
must  be  less  than  1 ;  that  is,  they  must  be  fractions.  Those 
to  the  numbers  between  10  and  100,  must  fall  between  1  and  2 ; 
that  is,  they  are  mixed  numbers,  consisting  of  one  and  some 
fraction.  Likewise  the  indices  to  the  numbers  between  100 
and  1000,  will  fall  between  2  and  3 ;  that  is,  they  are  mixed 
numbers,  consisting  of  2  and  some  fraction ;  and  so  of  the 
other  indices. 

Hereafter  the  integral  part  only  of  these  indices  will  be 
called  the  index ;  and  the  fractional  part  will  be  called  the 
logarithm.  The  computation  of  these  fractional  parts  is  called 
making  logarithms  ;  and  the  most  troublesome  part  of  this  work 
is  to  make  the  logarithms  of  prime  numbers,  or  those  which 
caiwot  be  divided  by  any  other  numbers  than  themselves  and 
unity. 

RULE 

For  computing  the  Logarithms  of  Numbers* 

Let  the  sum  of  its  proposed  number  and  the  next  less  num- 
ber be  called  A.  Divide  0.8685889638+  by  A,  and  reserve 

*  The  number  0.8685889638-}-  is  twice  the  reciprocal  of,  the  hyper- 
bolic log.  2.302585093,  which  is  the  log.  of  10,  according  to  the  first  form 
of  Lord  Napier,  the  inventor  of  logarithms ,;  which  log.  according  to  the 
excellent  Sir  I.  Newton's  method  is  calculated  thus ;  let  DFD  (PI.  14, 
jig.  1)  be  an  hyperbola  whose  centre  is  C,  vertex  F,  and  interposed 
square  CAFE=1.  In  CA  take  AB  and  Ab,  on  each  side  =  JL,  or  0.1 ; 
and,  erecting  the  perpendiculars  BD,  bd,  half  the  sum  of  the  spaces  AD  and 
Ad  will  be  =0.1  |  °7  1  *7'+™»™  &c. 
and  the  half  diff.  =o_°]+°^+o.°oo«,i  |  o.oooooom  &c< 

Which  reduced  will  stand  thus, 

0.1000000000000,0.0050000000000  Sum  of  these=0.1 053605 156577=Arf 
3333333333  250000000  And  the  diff.  =0.0953 101798043=AD 

20000000  1666666  In  like  manner  putting  AB  and  A6 

142857  12500  each  =  0.2  there  is  obtained 

1111  100  Ad  =  0.2231435513142,  and 

_9 J_AD  =  0.1823215567939. 

$1003353477310,0.0050251679267 

Having  thus  the  hyperbolic  logarithms  of  the  four  decimal  numbers  0.8, 
0.9, 1.1,  and  1.2 ;  and  since  ^X^=2,  and  0.8  and  0.9  are  less  than  unity, 
adding  their  logarithms  to  double  the  log.  of  1.2,  we  have  0.6931471805507, 
the  hyperbolic  log.  of  2.  To  the  triple  of  this  adding  the  log.  of  0.8,  because 
Hgl— 10,  we  have  2.3025850929933,  the  log.  of  10.  Hence  by  one  addition 


30  OF  LOGARITHMS. 

the  quotient.  Divide  the  reserved  quotient  by  the  square  of  A, 
and  reserve  this  quotient.  Divide  the  last  reserved  quotient 
by  the  square  of  A,  reserving  the  quotient  still ;  and  thus  pro- 
ceed as  long  as  division  can  be  made.  Write  the  reserved 
quotients  orderly  under  one  another,  the  first  being  uppermost. 
Divide  these  quotients  respectively  by  the  odd  numbers  1,  3, 
5,  7,  9,  11,  &c. ;  that  is,  divide  the  first  reserved  quotient  by  1, 
the  second  by  3,  the  third  by  5,  the  fourth  by  7,  &c.,  and  let 
these  quotients  be  written  orderly  under  one  another;  add  them 
together,  and  their  sum  will  be  a  logarithm.  To  this  logarithm 
add  the  logarithm  of  the  next  less  number,  and  the  sum  will 
be  the  logarithm  of  the  number  proposed. 

EXAMPLE    1. 

Required  the  logarithm  of  the  number  2. 
Here  the  next  less  number  is  1,  and  2+l=3=A,  and  A3 
or  32  =9 ;  then 
3)0.868588964 


9)0.289529654 -^   1=0.289529654 


9)0.032169962-^-  3=0.010723321 
9)0.003574440-r  5=0.000714888 


9)0.0003971 60-H  7=0.000056737 
9)0.000044129-f-  9=0.000004903. 
9)0.000004903-^  1 1  =0.000000446 


9)0.000000545-^-13=0.000000042 
0.000000061 -M5=0.000000004 


are  found  the  logarithms  of  9  and  11 :  And  thus  the  logarithms  of  all  the 
prime  numbers  are  prepared,  that  is,  2,  3,  5,  11,  &c. 

Moreover,  by  only  depressing  the  numbers  above  computed,  lower  in 
the  decimal  places,  and  adding,  are  obtained  the  logarithms  of  the  decimals 
0.98,0.99,  1.01,  1.02;  as  also  of  these,  0.998,  0.999,  1.001,  1.002.  And 
hence,  by  addition  and  subtraction,  will  arise  the.logarithms  of  the  primes 
7, 13, 17, 3T,  &c.  All  which  logarithms  being  divided  by  2.3025850929933 
<the  hyperbolic  tog.  of  10),  or  multiplied  by  its  reciprocal,  .4342944819, 
give  the  common  logarithms  to  be  inserted  in  the  table. 

Note. — For  further  illustration  on.  this  subject,  the  reader  is  referred  to 
Button's  Tables, 


OF  LOGARITHMS.  31 

To  this  logarithm  0.301029995 
add  the  logarithm  of  1=0.000000000 

Their  sum=0.30 1029995 =log.  of  2. 
The  manner  in  which  the  division  is  here  carried  on  may  be 
readily  perceived  by  dividing,  in  the  first  place,  the  given  deci- 
mal by  A,  and  the  succeeding  quotients  by  A2  ;  then  letting 
these  quotients  remain  in  then-  situation,  as  seen  in  the  example, 
divide  them  respectively  by  the  odd  numbers,  and  place  the  new 
quotients  in  a  column  by  themselves.  By  employing  this  pro- 
cess, the  operation  is  considerably  abbreviated. 

EXAMPLE    2. 

Required  the  logarithm  of  the  number  3. 
Here  the  next  less  number  is  2;  and3+2=5=A,andAa=25. 
5)0.868588964 


25)0.173717793-^  1=0.173717793 
25)0.006948712^-  3=0.002316237 
25)0.000277948  -r-  5=0.000055590 
25)0.00001I118-r  7=0.000001588 
25)0.000000445—  9=0.000000049 
0.000000018H- 1 1  =0.000000002 


To  this  logarithm   0.176091259 
add  the  logarithm  of  2=0.301029995 


Their  sum=0.477121254=log.  of  3. 
Then,  because  the  sum  of  the  logarithms  of  numbers  gives 
the  logarithm  of  their  product ;  and  the  difference  of  the  loga- 
rithms gives  the  logarithm  of  the  quotient  of  the  numbers : 
from  the  two  preceding  logarithms,  and  the  logarithm  of  10, 
which  is  1,  a  great  many  logarithms  can  be  easily  made,  as  in 
the.  folio  wing  examples. 

Example  3.     Required  the  logarithm  of  4. 

Since  4=2X2,  then  to  the  logarithm  of  2=0.301029995 

add  the  logarithm  of  2=0.301029995 

»  " 

The  sura=logarithm  of  4=0.602059990 


\  OF  LOGARITHMS. 

Example  4.     Required  the  logarithm  of  5. 
10-r2  being  =5,  therefore  from  the  logarithm  of 

10=1.000000000 
subtract  the  logarithm  of    2=0.301029995 


the  remainder  is  the  logarithm  of  5=0.698970005 

Example  5.     Required  the  logarithm  of  6. 

6=3X2,  therefore  to  the  logarithm  of  3=0.477121254 
add  the  logarithm  of  2=0.301029995 

their  sum  =  logarithm  of  6=0.778151249 

Example  6.     Required  the  logarithm  of  8. 
8=23,  therefore  multiply  the  logarithm  of  2=0.301029995 

by  3 

The  product  =  logarithm  of  8=0.903089985 

Example  7.    Required  the  logarithm  of  9. 

9=32,  therefore  the  logarithm  of  3=0.477121254 
being  multiplied  by  2 

the  product  =  logarithm  of  9=0.954242508 

Example  8*    Required  the  logarithm  of  7. 
Here  the  next  less  number  is  6,  and  7+ 6  =13= A,  and 
a  =  169. 

13)0.868588964 

. 

169)0.066814536-7-1=0.066814536 


169)0.000395352-f-3=0.000 1 31 784 

169)0.000002339-7-5=0.000000468 

0.000000014-7-7=0.000000002 


To  this  logarithm  =0.066946790 
add  the  logarithm  of  6=0.778151249 


Their  sum  —  0.845098039  =  logarithm  of  7. 


OF  LOGARITHMS. 


fof  12 
of  14 
of  J  5  is  equal  to  the  sum  of  the 
of  16             logarithms 
of  18 
of  20 

•of  3  and  4 
of  7  and  2. 
of  3  and  5. 
of  4  and  4. 
of  3  and  6. 
..  of  4  and  5. 

The  logarithm 


The  logarithms  of  the  prime  numbers  11, 13, 17, 19,  &c.  being 
computed  by  the  foregoing  general  rule,  the  logarithms  of  the 
intermediate  numbers  are  easily  found  by  composition  and  divi- 
sion. It  may  however  be  observed,  that  the  operation  is  shorter 
in  the  larger  prime  numbers ;  for  when  any  given  number  ex- 
ceeds 400,  the  first  quotient  being  added  to  the  logarithm  of  its 
next  lesser  number,  will  give  the  logarithm  sought,  true  to  eight 
or  nine  places ;  and  therefore  it  will  be  very  easy  to* examine  any 
suspected  logarithm  in  the  Tables. 

For  the  arrangement  of  logarithms  in  a  table,  the  method  of 
finding  the  logarithm  of  any  natural  number ,  and  of  finding  the 
natural  number  corresponding  to  any  given  logarithm  therein, — 
likewise  for  particular- rules  concerning  the  indices,  the  reader  will 
consult  Table  1,  with  its  explanation  at  the  end  of  this  treatise. 

MULTIPLICATION. 

Two  or  more  numbers  being  given,  to  find  their  product  by 
Logarithms. 

RULE. 

Having  found  the  logarithms  of  the  given  numbers  in  the 
table,  add  them  together,  and  their  sum  is  the  logarithm  of  the 
product ;  which  logarithm  being  found  in  the  table,  will  give  a 
natural  number,  that  is,  the  product  required. 

Whatever  is  carried  from  the  decimal  part  of  the  logarithm 
is  to  be  added  to  the  affirmative  indices,  but  subtracted  from 
the  negative.  Likewise  the  indices  must  be  added  together 
when  they  are  all  of  the  same  kind,  that  is,  when  they  are  all  affir- 
mative, or  all  negative ;  but  when  they  are  of  different  kinds, 
the  difference  must  be  found,  which  will  be  of  the  same  denomi- 
nation with  the  greater. 

Example  1.  Required  the  product  of  86.25  multiplied  by 
6.48. 

Log.  of  86.25  =  1.935759 
Log.  of    6.48=0.811575 


Product  =  558.9  =  2.747334* 

*  For  the  method  of  finding  the  natural  number,  answering  to  the  sum  of 
the  logarithms,  the  reader  will  consult  Table  1,  at  the  end  of  this  treatise, 

B3 


t  OF  LOGARITHMS. 

Example  2.    Required  the  product  of  46.75  and  .327& 
Log.  of  46.75  =     1.669782 
Log.  of  .3275  =—1,515211 


Product  =  15.31 +=1.184993 

Here  the  +1  that  is  to  he  carried  from  the  decimals  can* 
eels  the  — 1,  and  consequently  there  remains  1  in  the  upper  line 
.to  be  set  down. 

1    Example3.     Required  the  product  of  3.768»  2.053,,  and 
.007693. 

Log.  of      3.768=*     0.576111 

Log.  of      2,053  =     0.312389 

Log.  of  .007693  =—-3.886096 


Product  =  .05951+=— 2.774596 

In  this  example  there  is  1  to  carry  from  the  decimal  part 
of  the  logarithms,  which,  subtracted  from  — 3,  the  negative  in- 
dex, leaves  2,  the  index  of  the  sum  of  the  logarithms,  and 
is  negative. 

Example  4.  Required  the  product  of  27.63,  1.859,  .7258, 
and  0.3591. 

Log.  of  27.63=     1.441381 

Log.  of  1.859  =     0.269279 

Log.  of  .7258=— 1.860817 

Log.  of  .03591  =—2.555215 


Product  nearly  =  1.339=     0.126692 

In  this  example  there  is  2  to  carry  from  the  decimal  part  of 
the  logarithms,  which  added  to  1,  the  affirmative  index,  makes 
.3,  from  this  take  3,  the  sum  of  the  negative  indices,  the  re- 
mainder is  0,  which  is  the  index  of  the  sum  of  the  logarithms. 

5.  Required  the  product  of  23.14  and  50.62,.  by  logarithms. 

Ans.  117.1347 

6.  Required  the  product  of  3.12567,  .02868,  and  .12379,  by 
logarithms.  Ans.  .01109705 

7.  Required  the  product  of  .1508,  .0139,  and  75&9,  by  lo- 
garithms. Ans.  1.586553 

8.  Required  the  product  of  637.8  and  8&27,  by  logarithms. 

Ans.  56936.406 

9.  Required  the  product  of  14  and  8.45,  by  logarithms. 

Ans.  118.30 


OF  LOGARITHMS.  36 

DIVISION. 

Two  numbers  being  given,  to  find  how  many  times  one  is  eon 
tained  in  the  other  by  Logarithms. 

RULE. 

From  the  logarithm  of  the  dividend  subtract  the  logarithm 
of  the  divisor,  and  the  remainder  will  be  the  logarithm  whose 
corresponding  natural  number  will  be  the  quotient  required. 
;••  In  this  operation,  the  index  of  the  divisor  must  be  changed 
from  affirmative  to  negative,  or  from  negative  to  affirmative ; 
and  then  the  difference  of  the  affirmative  and  negative  indices 
must  be  taken  for  the  index  to  the  logarithm  of  the  quotient. 
Likewise  when  1  has  been  borrowed  in  the  left-hand  place 
of  the  decimal  part  of  the  logarithm,  add  it  to  the  index  of 
the  divisor,  if  affirmative ;  but  subtract  it,  if  negative ;  and  let 
the  index  thence  arising  be  changed  and  worked  with  as  before. 
Example  1.  Divide  558.9  by  6.48. 

Log.  of  558.9  =  2.747334 

Log.  of     6.48  =  0.811575 


Quotient  =  85.25=  1.935759 

Here,  the  1  to  be  taken  from  the  decimals  is  taken  as  — 1, 
which  when  added  to  2,  the  index  of  the  dividend,  leaves  1  for 
the  index  of  the  quotient;  that  is,  2 — 1=K 
Example  2.     Divide  15.31  by  46.75. 

Log.  of  15.31=         1.184975 
Log.  of  46.75  =          1.669782 


Quotient =.3275  =—1.515193 

Here,  the  1  to  be  taken  from  the  decimals  is  added  to  1,  the 
index  of  the  divisor  makes  2 ;  this  with  its  sign  changed  is  — 2, 
from  which  subtracting  1,  the  index  of  the  dividend,  the  re- 
mainder is — 1,  which  is  negative,  because  the  negative  index 
is  greater. 

Example  3.     Divide  .05951  by  .007693. 

Log.  of  .0595 1     =—  2.774590 
Log.  of  .007693  =—3.886096 

Quotient  =  7.735  =     0.888494 

Here,  the  1  to  be  taken  from  the  decimals  is  subtracted 
from  — 3,  which  leave  — 2,  this  changed  is  -f-2  ?  and  this 
added  to  — 2,  the  other  index,  gives  2 — 2=0. 


36  OF  LOGARITHMS, 

Example  4.    Divide  .6651  by  22.5. 

Log.  of  .6651  =— 1.822887 
Log.  of    22.5=      1.352183 

*  ___ 

Quotient  =  .02956  =  —2.470704 

Here,  +1  in  the  lower  index,  is  changed  into — 1,  and  this- 
added  to  — 1,  the  other  index,  gives  — 1 — 1,  or  — 2,  the  in- 
dex of  the  result. 

5v  Required  the  quotient  of  125  divided  by  1728,  by  loga- 
rithms- Ans.  .0723379 
6.  Divide  1728.95  by  1.10678,  by  logarithms. 

Ans.  1562.144 
•   7.  Divide  .067859  by  1234.59,  by  logarithms. 

Ans.  .0000549648 

8.  Divide  .7438  by  12.9470,  by  logarithms.     Ans,  .057449 

9,  Divide  .06314  by  .007241,  by  logarithms.    Ans.  8.71979 


PROPORTION, 

Or  the  Rule  of  Proportion  in  Logarithms* 

RULE. 

Having  stated  the  three  given  terms  according  to  the  rule  in 
common  Arithmetic,  write  them  orderly  under  one  another,  with 
the  signs  of  proportion ;.  then  add  the  logarithms  of  the  second 
and  third  terms  together,  and  from  their  sum  subtract  the  loga- 
rithm of  the  first  term,  and  the  remainder  will  be  the  loga- 
rithm of  the  fourth  term,  or  answer. 

Or, — add  together  the  arithmetical  complement  of  the  loga- 
rithm of  the  first  term,  and  the  logarithms  of  the  second  and 
third  terms  ;  the  sum,  rejecting  10  from  the  index,  will  be  the 
logarithm  of  the  fourth  term,  or  term  required. 

N.B.  The  arithmetical  complement  of  a  logarithm  is  what 
it  wants  of  10,000000,  or  20,000000,  and  the  easiest  way  to 
find  it  is  to  begin  at  the  left-hand,  and  subtract  every  figure  from 
9,  except  the  last,  which  should  be  taken  from  10 ;  but  if  the 
index  exceed  9,  it  must  be  taken  from  19. — It  is  frequently  used 
in  the  rule  of  Proportion  and  Trigonometrical  calculations,  to 
change  subtractions  into  additions.* 

*  When  the  index  is  negative  add  it  to  9,  and  subtract  as  before.  And 
for  every  arithmetical  complement  that  is  added,  subtract  10  from  the  last 
sum  of  the  indices. 


OF  LOGARITHMS.  37 

EXAMPLES. 

I  st.     If  a  clock  gain  1 4  seconds  in  5  days  1 8  hoursrhow  much 
will  it  gain  in  17  days  15  hdurs  ? 

5.75  days        :    Log.  =  0.759668 

17.625  days  : :    Log.  =  1.246129 
14  seconds      :    Log.  =  1.146128 

2.392257 


Answer  =42".  91  =  U632589 
Or  thus  ;  5.75  days :  Arith.  Co.  Log.  =  9.240332 
17,625 ::  Log.  =  1.246129 

14  seconds  :  Log.  =  1.146128 

Answer  =  42".91  =  1.632589 
3d.    Find  a  fourth  proportional  to  98.45,  1.969, 
98.45     :     Log.  =  1.99321 6 

347.2::     Log.  =  2.540580 
1.969     :     Log.  =  0.294246 

2.834826 


Answer  =  6.944  =  0.841610 

3d.    What  number  will  have  the  same  proportion  to  .8538 
as  .3275  has  to  .0131 1 

.0131     i    Log.  =—2.117271 

.3275  :  :     Log.  =—1.515211 

.8538    :     Log.  =—1.931356  ^ 

—1.446567 


Answer  =  21.35  =     1.329296 

4th    Required  a  third  proportional  number  to  9.642  and  4.821 
9.642     i    Log.  =  0.984 167 

4.821  :  :    Log.  =  0.683137 
4.821    :    Log.  =  0.683137 

1.366274 


Answer=  2.411  =  0.382107 


38  OF  LOGARITHMS. 

5.  Find  a  fourth  proportional  to  .05764,  .7186,  and  .34721, 
by  logarithms.  Ans.  4.328681. 

6.  Findafourth  proportional  to  12.687, 14.065,  and  100.979, 
by  logarithms.  Ans.  112.0263. 

7.  Find  a  mean  proportional  between  8.76  and  43.5,  by  loga- 
rithms. Ans.  16.7051. 

8.  Find  a  third  proportional  to  12.796   and  3.24718,  by 
logarithms.  Ans.  .8240216. 

9.  If  the  interest  of  £100  for  a  year,  or  365  days,  be  £4.5, 
wliat  will  be  the  interest  of  £279.25  for  274  days  ? 

Ans.  £9.433294. 


INVOLUTION. 

Tojind  any  proposed  power  of  a  given  number  by  Logarithms. 
RULE. 

Multiply  the  logarithm  of  the  given  number  by  the  index  of 
the  proposed  power,  and  the  product  will  be  the  logarithm 
whose  natural  number  is  the  power  required. 

When  a  negative  index  is  thus  multiplied,  its  product  is 
negative,  but  what  was  carried  from  the  decimal  part  of  the 
logarithm  must  be  affirmative ;  consequently  the  difference  is 
the  index  of  the  product,  which  difference  must  be  considered 
of  the  same  kind  with  the  greater,  or  that  which  was  made  the 
minuend. 

EXAMPLES. 

1.  What  is  the  second  power  of  3.874  ? 

Log.  of  3.874  =  0.588160 
^  Index  =  2 

'  i    . 


Power  required  =  15.01  =  1.176320 

2.  Required  the  third  power  of  the  number  2.768. 
Log.  of  2.768  =  0,442166 
Index  =  3 


Answer  =  21.21  =  1.326498 
3.  Required  the  second  power  of  the  number  .2857. 
Log.  of  .2857  =  — 1.455910 
Index  =s  2 


Answer^  .08162= — 2.911820 


OF  LOGARITHMS, 

4.  Required  the  third  power  of  the  number  .791.6* 
Log.  of  .7916  =  — 1.898506 
Index  •  =  3 


Answer  =  .4961  =  —1.695518 

Hence,  3  times  the  negative  index  being  — 3,  and  2  to  cany 
from  the  decimals,  the  difference  is  — 1,  the  index  of  the  product. 

5.  To  find  the  4th  power  of  .09163.          Ans.  .000070494. 

6.  To  find  the  2d  power  of  6.05987.  Ans.  36.72203". 

7.  To  find  the  cube  of  3.07146.  Ans.  28.97575. 

8.  To  find  the  7th  power  of  1.09684.  Ans.  1.909864. 

9.  To  find  the  365th  power  of  1.0045.          Ans.  5.148888. 

EVOLUTION. 

To  extract  any  proposed  Root  of  a  given  number  by  Logarithms. 

RULE. 

Find  the  logarithm  of  the  given  number,  amj  divide  it  by 
the  index  of  the  proposed  root ;  the  quotient  is  a  logarithm 
whose  natural  number  is  the  root  required. 

When  the  index  of  the  logarithm  to  be  divided  is  negative, 
and  does  not  exactly  contain  the  divisor  without  some  remainder, 
increase  the  index  by  such  a  number  as  will  make  it  exactly 
divisible  by  the  index,  carrying  the  units  borrowed  as  so  many 
tens  to  the  left-hand  place  of  the  decimal,  and  then  divide  as  in- 
whole  numbers. 

EXAMPLES. 

1.  Required  the  square  root  of  847. 
Index  2)2.927883  =  log.  of  847. 

1.463941  =  quot.  =  log.  of  29.103+=  Ans. 

2.  Required  the  cube  root  of  847. 

Index  3)2.927883  =  log.  of  the  given  number. 

0.975961  =  quot.  =  log.  of  9.462  =  Ans.  nearly. 

3.  Required  the  square  root  of  .093. 
Index  2)— 2.968483=  log.  of  .093. 

—1.484241  =  quot.  =  log.  of  .304959=  Ans. 

4.  Required  the  cube  root  of  12345. 
Index  3)4.09 1491  =  log.  of  12345. 

1.363830  =  quot.  =  log  of  23. 1 16  =  Ans. 


4ft  GEOMETRY. 

5.  To  find  the  cube  root  of  .00048. 

Power,  or  index  3)4. 68 124 12=  log.  of  the  number. 


Root  .07829735 2.8937471  =  log.  of  the  root. 

Here  the  divisor  3  not  being  exactly  contained  in  4,  augment 
it  by  2,  to  make  it  become  6,  in  which  the  divisor  is  contained 
just  2  times ;  and  the  2  borrowed  being  prefixed  to  the  other 
figures,  makes  2.68124 12,  which  divided  by  3  gives  .8937471 ; 
therefore,  2.8937471  is  the  log.  of  the  root. 

6.  To  find  the  fourth  root  of  .967845,  by  logarithms.     Ans. 
.9918624. 

7.  To  find  the  cube  root  of  2.987635.     Ans.  1.440265. 

8.  To  find  the  cube  root  of  37^-5.     Ans.  .6827842. 

9.  To  find  the  value  of  (.001234)^.     Ans.    .0115047. 
10.  To  find  the  tenth  root  of  2.     Ans.  1.07 1 773. 


SECTION  IV. 
ELEMENTS  OF  PLANE  GEOMETRY. 


DEFINITIONS. 
See  PLATE  I. 

1.  GEOMETRY  is  that  science  wherein  we  consider  the  prop- 
erties of  magnitude. 

2.  A  point  is  that  which  has  no  parts,  being  of  itself  indivisi- 
ble ;  as  A. 

3.  A  line  has   length  but   no  breadth ;  as  AB,  figures  1 
and  2. 

4.  The  extremities  of  a  line  are  points,  as  the  extremities  of 
the  line  AB  are  the  points  A  and  J3,  figures  1  and  2. 

5.  A  right  line  is  the  shortest  that  can  be  drawn  between 
any  two  points,  as  the  line  AB,  fig.  1 ;    but  if  it  be  not  the 
shortest,  it  is  then  called  a  curve  line,  as  AB,  fig.  2» 

6.  A  superficies  or  surface  is  considered  only  as  having 
length  and  breadth,  without  thickness,  as  ABCD,  fig.  3* 

7.  The  extremities  of  a  superficies  are  lines. 

8.  The  inclination  of  two  lines  meeting  one  another  (provided 
they  do  not  make  one  continued  line),  or  the  opening  between 
them,  is  called  an  angle.     Thus  in  fig.  4  the  inclination  of  the 


GEOMETRY.  41 

line  AB  to  the  line  BC,  meeting  each  other  in  the  point  B,  or 
the  opening  of  the  two  lines  BA  and  BC,  is  called  an  angle,  as 
ABC. 

Note. — When  an  angle  is  expressed  by  three  letters,  the 
middle  one  is  that  at  the  angular  point. 

9.  When  the  lines  that  form  the  angle  are  right  ones,  it  is 
then  called  a  right-lined  angle,  as  ABC,  fig.  4.     If  one  of  them 
be  right  and  the  other  curved,  it  is  called  a  mixed  angle,  as  B, 
fig.  5.     If  both  of  them  be  curved,  it  is  called  a  curved-lined  or 
spherical  angle,  as  C,  fig.  6. 

10.  If  a  right  line  CD  (fig.  7)  fall  upon  another  right  line 
AB,  so  as  to  incline  to  neither  side,  but  make  the  angles  ADC, 
CDB,  on  each  side  equal  to  each  other,  then  those  angles  are 
called  right  angles,  and  the  line  CD  a  perpendicular. 

11.  An  obtuse  angle  is  that  which  is  wider  or  greater  than 
a  right  one,  as  the  angle  ADR,  fig.  7,  and  an  acute  angle  is 
less  than  a  right  one,  as  EDB,  fig.  7. 

12.  Acute  and  obtuse  angles  in  general  are  called  oblique 
angles. 

13.  If  a  right  line  CB,  fig.  8,  be  fastened  at  the  end  C,  and 
the  other  end  B  be  carried  quite  round,  then  the  space  compre- 
hended is  called  a  circle ;  and  the  curve  line  described  by  the 
point  B  is  called  the  circumference  or  the  periphery  of  the 
circle ;  the  fixed  point  C  is  called  its  centre. 

14.  The  describing  line  CZ?,  fig.  8,  is  called  the  semidiam- 
eter  or  radius ;  so  is  any  line  from  the  centre  to  the  circum- 
ference ;  whence  all  radii  of  the  same  or  of  equal  circles  are 
equal. 

1 5.  The  diameter  of  a  circle  is  a  right  line  drawn  through 
the  centre,  and  terminating  in  opposite  points  of  the  circum- 
ference ;  and  it  divides  the  circle  and  circumference  into  two 
equal  parts,  called  semicircles ;  and  is  double  the  radius,  as 
AB  or  DE,  fig.  8. 

16.  The  circumference  of  every  circle  is  supposed  to  be 
divided  into  360  equal  parts  called  degrees,  and  each  degree  into 
60  equal  parts  called  minutes,  and  each  minute  into  60  equal 
parts  called  seconds,  and  these  into  thirds,  fourths,  &c.  these 
parts  being  greater  or  less  as  the  radius  is. 

17.  A  chord  is  a  right  line  drawn  from  one  end  of  an  arc  or 
arch  (that  is,  any  part  of  the  circumference  of  a  circle)  to  the 
other,  and  is  the  measure  of  the  arc.     Thus  the  right  line  HG 
is  the  measure  of  the  arc  HBG+  fig  8. 

18.  The  segment  of  a  circle  is  any  part  thereof  which  is  cut 
off  by  a  chord  :  thus  the  space  which  is  comprehended  between 
the  chord  HG  and  the  arc  HBG,  or  that  which  is  compre* 


'42  GEOMETRY. 

hended  between  the  said  chord  HG  and  the  arc  HDAEG  are 
called  segments.     Whence  it  is  plain,  fig.  8, 

1.  That  any  chord  will  divide  the  circle  into  two  segments. 

2.  The  less  the  chord  is,  the  more  unequal  are  the  segments. 

3.  When  the  chord  is  greatest  it  becomes  a  diameter,  and 
then  the  segments  are  equal ;    and  each  segment  is  a  semi- 
circle.* 

19.  A  sector  of  a  circle  is  a  part  thereof  less  than  a  semi- 
circle, which  is  contained  between  two  radii  and  an  arc  :  thus 
the  space  contained  between  the  two  radii  CH,  CB,  and  the 
arc  HB  is  a  sector,  fig.  8. 

20.  The  right  sine  of  an  arc  is  a  perpendicular  line  let  fall 
from  one  end  thereof,  to  a  diameter  drawn  to  the  other  end : 
thus  HL  is  the  right  sine  of  the  arc  HB. 

The  sines  on  the  same  diameter  increase  till  they  come  to 
the  centre,  and  so  become  the  radius  ;  hence  it  is  plain  that  the 
radius  CD  is  the  greatest  possible  sine,  and  thence  is  called 
the  whole  sine. 

Since  the  whole  sine  CD  (fig.  8)  must  be  perpendicular  to 
the  diameter  (by  def.  20),  therefore  producing  DC  to  E,  the 
two  diameters  AB  and  DE  cross  one  another  at  right  angles, 
and  thus  the  periphery  is  divided  into  four  equal  parts,  as  BD, 
DA,  AE,  and  EB  (by  def.  10) ;  and  so  BD  becomes  a  quad- 
rant, or  the  fourth  part  of  the  periphery ;  therefore  the  radius  DC 
is  always  the  sine  of  a  quadrant,  or  of  the  fourth  part  of  the 
circle  BJ). 

Sines  are  said  to  be  of  as  many  degrees  as  the  arc  contains 
parts  of  360  :  so  the  radius  being  the  sine  of  a  quadrant  becomes 
the  sine  of  90  degrees,  or  the  fourth  part  of  the  circle,  which  is 
360  degrees. 

21.  The  versed  sine  of  an  arc  is  that  part  of  the  diameter 
that  lies  between  the  right  sine  and  the  circumference  :  thus 
LB  is  the  versed  sine  of  the  arc  HB,  fig.  8. 

22.  The  tangent  of  an  arc  is  a  right  line  touching  the  peri- 
phery, being  perpendicular  to  the  end  of  the  diameter,  and  is 
terminated  by  a  line  drawn  from  the  centre  through  the  other 
end :  thus  BK  is  the  tangent  of  the  arc  HB,  fig.  8. 

23.  And  the  line  which  terminates  the  tangent,  that  is,  CK, 
is  called  the  secant  of  the  arc  HB,  fig.  8. 

24.  What  an  arc  wants  of  a  quadrant  is  caljed  the  comple- 
ment thereof:  thus  D7/is  the  complement  of  th/are  #.#,  fig.  8. 

25.  And  what  an  arc  wants  of  a  semicircle  is  called  the  sup- 

*  For  the  demonstration  of  this  consult  Prop.  15,  Book  III.  Simpson's 
Euclid, 


GEOMETRY.  43 

plement  thereof:  thus  AH  is  the  supplement  of  the  arc  HB9 
fig.  8. 

26.  The  sine,  tangent,  or  secant  of  the  complement  of  any 
arc  is  called  the  co-sine,  co-tangent,  or  co-secant  of  the  arc 
itself:  thus  FH  is  the  sine,  DI  the  tangent,  and  CI  the  secant 
of  the  arc  DH :  or  they  are  the  co-sine,  co-tangent,  or  co-secant 
of  the  arc  HB,  fig.  8. 

27.  The  sine  of  the  supplement  of  an  arc  is  the  same  with 
the  sine  of  the  arc  itself;  for  drawing  them  according  to  de£ 
30,  there  results  the  self-same  line :  thus  HL  is  the  sine  of  the 
arc  HB,  or  of  its  supplement  ADH,  fig.  8. 

28.  The  measure  of  a  right-lined  angle  is  the  arc  of  a  circle 
swept  from  the  angular  point,  and  contained  between  the  two 
lines  that  form  the   angle  :  thus  the  angle  HCB,  fig.  8,  ia 
measured  by  the  arc  HB,  and  is  said  to  contain  sot  many  de- 
grees as  the  arc  HB  does  ;  so  if  the  arc  HB  is  60  degrees,  the 
angle  HCB  is  an  angle  of  60  degrees. 

Hence  angles  are  greater  or  less  according  as  the  arc 
described  about  the  angular  point,  and  terminated  by  the  two 
sides,  contains  a  greater  or  less  number  of  degrees  of  the  whole 
circle. 

29.  The  sine,  tangent,  and  secant  of  an  arc  is  also  the  sine, 
tangent,  and  secant  of  an  angle  whose  measure  the  arc  is  ;  thus, 
because  the  arc  HB  is  the  measure  of  the  angle  HCB,  and 
since  HL  is  the  sine,  BK  the  tangent,  and  CK  the  secant,  BL 
the  versed  sine,  HF  the  co-sine,  Dl'the  co-tangent,  and  CI  the 
co-secant,  &c.  of  the  arc  BH ;  then  HL  is  called  the  sine, 
BK  the  tangent,  CK  the  secant,  &c.  of  the  angle  HCB,  whose 
measure  is  the  arc  HB,  fig.  8. 

30.  Parallel  lines  are  such  as   are  equidistant  from  each 
other,  as  AB,  CD,  fig.  9. 

31.  A  figure  is  a  space  bounded  by  a  line  or  lines.     If  the 
lines  be  right  it  is  called  a  rectilineal  figure ;  if  curved  it  is 
called  a  curvilineal  figure  ;  but  if  they  be  partly  right  and  partly 
curved  lines  it  is  called  a  mixed  figure. 

32.  The  most  simple  rectilineal  figure  is  a  triangle,  being 
composed  of  three  right  lines,  and  is  considered  in  a  double 
capacity  :  1st,  with  respect  to  its  sides ;  and  2d,  to  its  angles. 

33.  In  respect  to  its  sides,  it  is  either  equilateral,  having  the 
three  sides  equal,  as  A,  fig.  10. 

34.  Or  isosceles,  having  two  equal  sides,  as  B,  fig.  11. 

35.  Or  scalene,  having  the  three  sides  unequal,  as  C,  fig.  12. 

36.  In  respect ^o  its  angles,  it  is  either  right-angled,  having 
one  right  angle,  as  D,  fig.  1 3, 

37.  Or  obiuse-angled,  having  one  obtuse  angle,  as  JS,  fig.  14 


44  GEOMETRY. 

38.  Or  acute-angled,  having  all   the  angles  acute,  as  JP, 
fig.  15. 

39.  Acute  and  obtuse-angled  triangles  are  in  general  called 
oblique-angled  triangles,  in  all  which  any  side  may  be  called 
the  base,  and  the  other  two  the  sides. 

40.  The  perpendicular  height  of  a  triangle  is  a  line  drawn 
from  the  vertex  to  the  base  perpendicularly  :  thus  if  the  triangle 
ABC  be  proposed,  and  BC  be  made  its  base,  then  if  from  the 
vertex  A  the  perpendicular  AD  be  drawn  to  BC,  the  line  AD 
will  be  the  height  of  the  triangle  ABC,  standing  on  BC  as  its 
base,  fig.  16. 

Hence  all  triangles  between  the  same  parallels  have  the 
same  height,  since  all  the  perpendiculars  are  equal  from  the 
nature  of  parallels. 

41.  Any  figure  of  four  sides  is  called  a  quadrilateral  figure. 

42.  Quadrilateral  figures,  whose  opposite  sides  are  parallel, 
are  called  parallelograms  :  thus  ABCD  is  a  parallelogram, 
fig.  3,  17,  and  AB,  fig.  18,  19. 

43.  A  parallelogram  whose  sides  are  all  equal  and  angles 
right  is  called  a  square,  as  ABCD,  fig.  17. 

44.  A  parallelogram  whose  opposite  sides  are  equal  and 
angles  right  is  called  a  rectangle,  or  an  oblong,  as  ABCD, 
fig.  3. 

45.  .A  rhombus  is  a  parallelogram  of  equal  sides,  and  has  its 
angles  oblique,  as  A,  fig.  18,  and  is  an  inclined  square. 

46.  A  rhomboides  is  a  parallelogram  whose  opposite  sides 
are  equal  and  angles  oblique ;  as  B,  fig.  19,  and  may  be  con- 
ceived as  an  inclined  rectangle. 

47.  Any  quadrilateral  figure  that  is  not  a  parallelogram  is 
called  a  trapezium.     Plate  7,  fig.  3. 

48.  Figures  which  consist  of  more  than  four  sides  are  called 
polygons ;  if  the  sides  are  all  equal  to  each  other,  they  are 
called  regular  polygons.     They  sometimes  are  named  from  the 
number  of  their  sides,  as  a  five-sided  figure  is  called  a  penta- 
gon, one  of  six  sides  a  hexagon,  <fec. ;  but  if  their  sides  are  not 
equal  to  each  other,  then  they  are  called  irregular  polygons,  as 
an  irregular  pentagon,  hexagon,  <fec. 

49.  Four  quantities  are  said  to  be  in  proportion  when  the 
product  of  the  extremes  is  equal  to  that  of  the  means  :  thus  if 
A  multiplied  by  D  be  equal  to  B  multiplied  by  C,  then  A  is 
said  to  be  to  B  as  C  is  to  D. 

POSTULATES,  OR  PETITIONS. 
1.  That  a  right  line  may  be  drawn  from  any  one  given  point 
to  another. 


GEOMETRY.  45 

2.  That  a  right  line  may  be  produced  or  continued  at  pleasure. 

3.  That  from  any  centre  and  with  any  radius  the  circum- 
ference of  a  circle  may  be  described. 

4.  It  is  also  required  that  the  equality  of  lines  and  angles  to 
others  given,  be  granted  as  possible  :  that  it  is  possible  for  one 
right  line  to  be  perpendicular  to  another  at  a  given  point  or  dis- 
tance ;  and  that  every  magnitude  has  its  half,  third,  fourth,  &c. 
part. 

Note. — Though  these  postulates  are  not  always  quoted,  the 
reader  will  easily  perceive  where  and  in  what  sense  they  are, 
to  be  understood. 


AXIOMS,  OR  SELF-EVIDENT  TRUTHS. 

1.  Things  that  are  equal  to  one  and  the  same  thing  are 
equal  to  each  other. 

2.  Every  whole  is  greater  than  its  part. 

3.  Every  whole  is  equal  to  all  its  parts  taken  together. 

4.  If  to  equal  things  equal  things  be  added,  the  whole  will 
be  equal. 

5.  If  from  equal  things  equal  things  be  deducted,  the  remain  • 
ders  will  be  equal. 

6.  If  to  or  from  unequal  things  equal  things  be  added  or 
taken,  the  sums  or  remainders  will  be  unequal. 

7.  All  right  angles  are  equal  to  one  another. 

8.  If  two  right  lines  not  parallel  be  produced  towards  their 
nearest  distance,  they  will  intersect  each  other. 

9.  Things  which  mutually  agree  with  each  other  are  equal. 

NOTES. 

A  theorem  is  a  proposition  wherein  something  is  proposed 
to  be  demonstrated. 

A  problem  is  a  proposition  wherein  something  is  to  be  done 
or  effected. 

A  lemma  is  some  demonstration  previous  and  necessary,  to 
render  what  follows  the  more  easy. 

A  corollary  is  a  consequent  truth,  deduced  from  a  foregoing 
demonstration. 

A  scholium  is  a  remark  or  observation  made  upon  something 
going  before. 


46  GEOMETRICAL 


GEOMETRICAL  THEOREMS. 

THEOREM  I. 
PL.  I.  Jig.  20. 

If  a  right  line  falls  on  another,  as  AB,  or  EB,  does  on  CD,  it  either 
makes  with  it  two  right  angles,  or  two  angles  equal  to  two  right  angles. 

1.  If  AB  be  perpendicular  to  CD,  then  (by  def.  10)  the  an- 
gles CBA  and  ABD  will  be  each  a  right  angle. 

2.  But  if  the  line  fall  slantwise,  as  EB,  and  let  AB  be  per- 
pendicular to   CD-,   then   the   Z.DBA=DBE-\-EBA  :    add 
ABC  to  each;   then,  DBA+ABC=DBE+EBA+ABC', 
but  CBE=EBA+ABC,  therefore  the  angles  DBE+EBC= 
DBA+ABC,  or  two  right  angles.     Q.  E.  D. 

Corollary  1.  Whence  if  any  number  of  right  lines  were 
drawn  from  one  point,  on  the  same  side  of  a  right  line,  all 
the  angles  made  by  these  lines  will  be  equal  to  two  right  angles. 

2.  And  all  the  angles  which  can  be  made  about  a  point  will 
be  equal  to  four  right  angles. 

THEOREM  II. 

PL.  I.  fig.  21. 

If  one  right  line  cross  another  (as  AC  does  BD\  the  opposite  angles 
made  by  those  lines  will  be  equal  to  each  other  :  that  is,  AEB  to  CED,  and 
BECtoAED. 

By  theorem  1,  BEC+CED=  two  right  angles. 
and  CED+DEA=  two  right  angles. 

Therefore  (by  axiom  1)  BEC+CED=CED+DEA; 
take  CED  from  both,  and  there  remains  BEC=DEA  (by 
axiom  5).  Q.  E.  D. 

After  the  same  manner  CED-{-AED=  two  right  angles  ;  and 
AED-}-AEB=  two  right  angles  ;  wherefore  taking  AED  from 
both,  there  remains  CED=AEB.  Q.  E.  D. 

THEOREM  III. 

PL.  M£-.  22. 

If  a  right  line  cross  two  parallels,  as  GH  docs  AB  and  CD,  then,  * 

1.  Their  external  angles  are  equal  to  each  other,  that  is,  GEB=CFH. 

2.  The  alternate  angles  will  be  equal,  that  is,  AEF=EFD  and  BEF 
—CFE.       • 

3.  The  external  angle  will  be  equal  to  the  internal  and  opposite  one  on  the 
same  side,  that  is,  GEB=EFD  and  AEG=CFE. 

4.  And  the  sum  of  the  internal  angles  on  the  same  side  are  equal  to  two 
right  angles  ;  that  is,  BEF-\-DFE  are  equal  to  two  right  angles,  and, 
AEF-\-CFE  are  equal  to  two  right  angles. 


THEOREMS.  47 

1.  Since  AB  is  parallel  to  CD,  they  may  be  considered  as 
one  broad  line,  crossed  by  another  line,  as  GH ;  then  (by  the 
last  theo.)  GEB=CFH,  and  AEG=HFD. 

2.  Also  GEB=AEF,  and  CFH=EFD ;  but  GEB=CFH 
(by  part  1.  of  this  theo.),  therefore  AEF=EFD.     The  same 
way  we  prove  FEB^EFC. 

3.  AEF=EFD  (by  the  last  part  of  this  theo.)  ;  but  AEF 
=  GEB  (by  theo.  2),  therefore  GEB=EFD.     The  same  way 
we  prove  AEG=CFE. 

4.  For  since  GEB=EFD,  to  both  add  FEE ;  then  (by 
axiom  4)  GEB+FEB=EFD+FEB ;  but  GEB+FEB  are 
equal  to  two  right  angles  (by  theo.  1),  therefore  EFD+FEB 
are  equal  to  two  right  angles  :  after  the  same  manner  we  prove 
that  AEF+  CFE  are  equal  to  two  right  angles.     Q.  E.  D.* 

THEOREM  IV. 

PL.  I.  Jig.  23. 

In  any  triangle  ABC,  one  of  its  legs,  as  jBC,  being  produced  towards  Dt 
it  will  make  the  external  angle  A  CD  equal  to  the  two  internal  opposite  an- 
gles taken  together ;  viz.  to  B  and  A. 

Through  C,  let  CE  be  drawn  parallel  to  AB;  then  since 
BD  cuts  the  two  parallel  lines  BA,  CE,  the  angle  ECD=B 
(by  part  3  of  the  last  theo.);  and  again,  since  AC  cuts  the 
same  parallels,  the  angle  ACE=A  (by  part  2  of  the  last), 
therefore  ECD+ACE=ACD=B+A.  Q.  E.  D. 

Cor.  1.  Hence,  if  a  triangle  have  its  exterior  angle  and  one 
of  its  opposite  interior  angles  double  of  those  in  another  tri- 
angle, its  remaining  opposite  interior  angle  will  also  be  double 
of  the  corresponding  angle  in  the  other,  f 

That  invaluable  instrument,  Hadley's  Quadrant,  is  founded  on 
this  corollary,  annexed  as  an  obvious  consequence  of  the  the- 
orem. A  ray  of  light  SA  (PL  14.  fg.  2)  from  the  sun, 
against  the  mirror  at  JL,  is  reflected  at  an  angle  equal  to  its  in- 
cidence ;  and  now  striking  the  half-silvered  glass  at  C,  it  is 
again  reflected  to  £,  where  the  eye  likewise  receives,  through 
the  transparent  part  of  that  glass,  a  direct  ray  from  the  boun- 
dary of  the  horizon. 

Hence,  the  triangle  AEC  has  its  exterior  angle  ECD  and 
one  of  its  interior  angles  CAE  respectively  double  of  the  ex- 
terior angle  BCD  and  the  interior  angle  CAB  of  the  triangle 

*  For  an  excellent  demonstration  of  this  theorem  (by  the  motion  of 
the  straight  line  crossing  the  parallel  lines  about  a  point  in  one  of  them), 
the  reader  will.consult  Leslie's  Geometry,  Prop.  23,  page  26. 

f  This  corollary,  with  the  following  demonstration,  is  found  in  Leslie's 
Geometry,  pages  32  and  406. 


V 


48  GEOMETRICAL 

ABC;  wherefore  the  remaining  interior  angle  AEC,  or  SEZ, 
is  double  of  ABC ;  that  is,  the  altitude  of  the  sun  above  the 
horizon  is  double  of  the  inclination  of  the  two  mirrors.  But 
the  glass  at  C  remaining  fixed,  the  mirror  at  A  is  attached  to 
a  moveable  index,  which  marks  their  inclination. 

The  same  instrument,  in  its  most  improved  state,  and  fitted 
with  a  telescope,  forms  the  sextant,  which,  being  admirably 
calculated  for  measuring  angles  in  general,  has  rendered  the 
most  important  services  to  geography  and  navigation. 

THEOREM  V. 
PL.  1.  jig.  23. 

In  any  triangle  ABC,  all  the  three  angles,  taken  together,  are  equal  to 
two  right  angles,  viz.  A-\-B-^-A  CB=  two  right  angles. 

Produce  CB  to  any  distance,  as  D,  then  (by  the  last)  ACD 
=J5+A;  to  both  add  ACB;  then  ACD+ACB=A+B+ 

ACB ;  but  ACD+ACB=  two  right  angles  (by  theo.  1) ;  there- 
fore the  three  angles  A+ B-\-  ACB=  two  right  angles.  Q.  E.  D. 

Cor.  1.  Hence  if  one  angle  of  a  triangle  be  known,  the  sum 
of  the  other  two  is  also  known  ;  for  since  the  three  angles  of 
every  triangle  contain  two  right  ones,  or  180  degrees,  therefore 
180 —  the  given  angle  will  be  equal  to  the  sum  of  the  other 
two ;  or  1 80 —  the  sum  of  two  given  angles  gives  the  other  one. 

Cor.  2.  In  every  right-angled  triangle,  the  two  acute  angles 
are  =  90  degrees,  or  to  one  right  angle;  therefore  90 —  one 
acute  angle  gives  the  other. 

THEOREM  VI. 
PL.  1.^.24. 

If  in  any  two  triangles,  ABC,  DEF,  there  be  two  sides  AB,  AC  in  the 
one  severally  equal  to  DE,  DF  in  the  other,  and,  the  angle  A  contained  be- 
tween the  two  sides  in  the  one  equal  to  D  in  the  other  ;  then  the  remaining 
angles  of  the  one  will  be  severally  equal  to  those*  of  the  other,  viz.  B=E, 
and  C=F;  and  the  base  of  the  one  EC  will  be  equal  to  EF,  that  of  the 
other. 

If  the  triangle  ABC  be  supposed  to  be  laid  on  the  triangle 
DEF,  so  as  to  make  the  points  A  and  B  coincide  with  D  and 
E,  which  they  will  do,  because  AB=DE  (by  the  hypothesis) ; 
and  since  the  angle  A=D,  the  line  AC  will  fall  along  DF, 
and  inasmuch  as  they  are  supposed  equal,  C  will  fall  in  F ; 
seeing  therefore  the  three  points  of  one  coincide  with  those 
of  the  other  triangle,  they  are  manifestly  equal  to  each 
other;  therefore  the  angle  #=£,  and  C=JF,  and  BC=EF. 
Q.  E.  D. 


THEOREMS.  49 

LEMMA. 
PL.  I.  Jig.  11. 

If  two  sides  of  a  triangle  abc  be  equal  to  each  other,  that  is,  ac=cb,  the 
angles  which  are  opposite  to  those  equal  sides  mil  also  be  equal  to  each 
other  ;  viz.  a—b.  , 

For  let  the  triangle  abc  be  divided  into  two  triangles  acd, 
deb,  by  making  the  angle  acd=dcb  (by  postulate  4) ;  then 
because  ac=bc,  and  cd  common  (by  the  last),  the  triangle 
adc=dcb  ;  and  therefore  the  angle  a=b.  Q.  E.  D. 

Cor.  Hence  if  from  any  point  in  a  perpendicular  which  bi- 
sects a  given  line  there  be  drawn  right  lines  to  the  extremities  of 
the  given  one,  they  with  it  will  form  an  isosceles  triangle. 

THEOREM  VII. 

PL.  I.  Jig.  25. 

The]  angle  BCD  at  the  centre  of  a  circle  ABED  is  double  the  angle 
BAD  at  the  circumference,  standing  upon  the  same  arc  BED. 

Through  the  point  A,  and  the  centre  C,  draw  the  line  A  CE ; 
then  the  angle  ECD=CAD+CDA  (by  theo.  4) ;  but  since 
AC=CD,  being  radii  of  the  same  circle,  it  is  plain  (by  the 
preceding  lemma)  that  the  angles  subtended  by  them  will  be 
also  equal,  and  that  their  sum  is  double  to  either  of  them,  that 
is,  DAC+ADC  is  double  to  CAD,  and  therefore  ECD  is 
double  to  CAD ;  after  the  same  manner  BCE  is  double  to 
CAB,  wherefore  BCE+ECD,  or  BCD,  is  double  to  BAG 
+  CAD,  or  to  BAD.  Q.  E.  D. 

Cor.  1 .  Hence  an  angle  at  the  circumference  is  measured 
by  half  the  arc  it  subtends  or  stands  on. 

Fig.  26. 

Cor.  2.  Hence  all  angles  at  the  circumference  of  a  circle 
which  stand  on  the  same  chord  as  AB  are  equal  to  each 
other,  for  they  are  all  measured  by  half  the  arc  they  stand  on, 
viz.  by  half  the  arc  AB 

Fig.  26. 

Cor.  3.  Hence  an  angle  in  a  segment  greater  than  a  semi- 
circle is  less  than  a  right  angle ;  thus  ADB  is  measured  by 
half  the  arc  AB ;  but  as  the  arc  AB  is  less  than  a  semicircle, 
therefore  half  the  arc  AB,  or  the  angle  ADB,  is  less  than  half 
a  semicircle,  and  consequently  less  than  a  right  angle. 

Fig.  27. 

Cor.  4.  An  angle  in  a  segment  less  than  a  semicircle  is  greater 
than  a  right  angle  ;  for  since  the  arc  AEC  is  greater  than  a 
semicircle,  its  half,  which  is  the  measure  of  the  angle  ABC, 

C 


50  GEOMETRICAL 

must  be  greater  than  half  a  semicircle,  that  is,  greater  than  a 
right  angle. 

Fig.  28. 

Cor.  5.  An  angle  in  a  semicircle  is  a  right  angle,  for  the  mea- 
sure of  the  angle  ABD  is  half  of  a  semicircle  AED,  and 
therefore  a  right  angle. 

THEOREM  VIII. 
PL.  1.  fig.  29. 

If  from  the  centre  C  of  a  circle  ABE  there  le  let  fall  the  perpendicular 
CD  on  the  chord  AB,  it  will  bisect  it  in  the  point  D. 

Let  the  lines  A  C  and  CB  be  drawn  from  the  centre  to  the 
extremities  of  the  chord  ;  then  since  CA=  CB,  the  angles  CAB 
=  CBA  (by  the  lemma).  But  the  triangles  ADC,  BDC  are 
right-angled  ones,  since  the  line  CD  is  a  perpendicular ;  and 
so  the  angle  ACD=DCB  (by  cor.  2,theo.  5);  then  have  we 
AC,  CD,  and  the  angle  AC  D  in  one  triangle  severally  equal 
to  CB,  CD,  and  the  angle  BCD  in  the  other;  therefore  (by 
theo.  6)  AD=DB.  Q.  E.  D. 

Cor.  Hence  it  follows,  that  any  line  bisecting  a  chord  at 
right  angles  is  a  diameter;  for  a  line  drawn  from  the  centre 
perpendicular  to  a  chord  bisects  that  chord  at  right  angles ; 
therefore,  conversely,  a  line  bisecting  a  chord  at  right  angles 
must  pass  through  the  centre,  and  consequently  be  a  diameter. 

THEOREM  IX. 
PL.  I.  fig.  29. 

If  from  the  centre  of  a  circle  ABE  there  be  drawn  a  perpendicular  CD  on 
the  chord  AB,  and  produced  till  it  meets  the  circle  in  F,  that  line  CF  will 
bisect  the  arc  AB  in  the  point  F. 

Let  the  lines  AF  and  BF  be  drawn  ;  then  in  the  triangles 
ADF,  BDF,  AD=BD  (by  the  last) ;  DF  is  common,  and 
the  angle  ADF=BDF,  being  both  right,  for  CD  or  DF  is  a 
perpendicular.  Therefore  (by  theo.  6)  AF=FB ;  but  in  the 
same  circle,  equal  lines  are  chords  of  equal  arcs,  since  they 
measure  them  (by  def.  19) ;  whence  the  arc  AF=FB,  and  so 
A FB  is  bisected  in  F  by  the  line  CF. 

Cor.  Hence  the  sine  of  an  arc  is  half  the  chord  of  twice 
that  arc.  For  AD  is  t)ie  sine  of  the  arc  AF  (by  def.  20), 
AF  is  half  the  arc,  and  AD  half  the  chord  AB  (by  theo.  83 ; 
therefore  the  corollary  is  plain. 


THEOREMS.  51 

THEOREM  X. 

PL.  1.^.30. 
In  any  triangle  ABD,  the  half  of  each  side  is  the  sine  of  the  opposite  angle. 

Let  the  circle  ADB  be  drawn  through  the  points  A,  B,  D ; 
then  the  angle  DAB  is  measured  by  half  the  arc  BKD  (by 
cor.  1,  theo.  7),  viz.  the  arc  BK  is  the  measure  of  the  angle 
BAD  -,  therefore  (by  cor.  to  the  last)  BE,  the  half  of  BD,  is 
the  sine  of  BAD :  in  the  same  way  may  be  proved  that  half  of 
AD  is  the  sine  of  ABD,  and  the  half  of  AB  the  sine  of  ADB. 
Q.  E.  D. 

THEOREM  XI. 
PL.  1.^.22. 

If  a  right  line  GH  cut  two  other  right  lines  AB,  CD,  so  as  to  make 
the  alternate  angles  AEF,  EFD  equal  to  each  other,  then  the  lines  AB  and 
CD  will  be  parallel. 

If  it  be  denied  that  AB  is  parallel  to  CD,  let  IK  be  parallel 
to  it ;  then  IEF=(EFD)=AEF  (by  part  2,  theo.  3),  a  greater 
to  a  less,  which  is  absurd,  whence  IK  is  not  parallel ;  and  the 
like  we  can  prove  of  all  other  lines  but  AB ;  therefore  AB  is 
parallel  to  CD.  Q.  E.  D. 

THEOREM  XII. 
PL.  \.fig.  3. 

If  two  equal  and  parallel  lines  AB,  CD,  be  joined  by  two  other  lines  AD, 
BC,  those  shall  be  also  equal  and  parallel. 

Let  the  diameter  or  diagonal  BD  be  drawn,  and  we  will  have 
the  triangles  ABD,  CBD,  whereof  AB  in  one  is  =  to  CD  in 
the  other,  BD  common  to  both,  and  the  angle  ABD=CDB 
(by  part  2,  theo.  3) ;  therefore  (by  theo.  6)  AD=CB,  and  the 
angle  CBD=ADB ;  and  thence  the  lines  AD  and  BC  are 
parallel,  by  the  preceding  theorem. 

Cor.  1.  Hence  the  quadrilateral  figure  ABCD  is  a  paral- 
lelogram, and  the  diagonal  BD  bisects  the  same,  inasmuch  as 
the  triangle  ABD— BCD,  as  now  proved. 

Cor.  2.  Hence  also  the  triangle  ABD  on  the  same  base 
AB,  and  between  the  same  parallels  with  the  parallelogram 
ABCD,  is  half  the  parallelogram. 

Cor.  3.  It  is  hence  also  plain  that  the  opposite  sides  of  a 
parallelogram  are  equal ;  for  it  has  been  proved  that,  ABCD 
being  a  parallelogram,  AB  will  be  =  CD,  and  AD=BC. 

C  2 


52  GEOMETRICAL 

THEOREM  XIII. 
PL.  I.  Jig.  91. 

All  parallelograms  on  the  same  or  equal  bases  and  between  the  same  par' 
allels  are  equal  to  one  another ;  that  is,  if  BD=GH,  and  the  lines  BH  and 
AF are  parallel,  then  the  parallelogram  ABDC=BDFE=EFHG. 

For  AC=BD=EF  (by  cor.  the  last) ;  to  both  add  CE, 
then  AE=CF.  In  the  triangles  ABE,  CDF,  AB=CD  and 
AE=CF,  and  the  angle  BAE=DCF  (by  part  3,  theo.  3); 
therefore  the  triangle  ABE=  CDF  (by  theo.  6) ;  let  the  triangle 
CKE  be  taken  from  both,  and  we  will  have  the  trapezium 
ABKC=KDFE ;  to  each  of  these  add  the  triangle  BKD, 
then  the  parallelogram  ABCD=BDEF:  in  like  manner  we 
may  prove  the  parallelogram  EFGH=BDEF.  Wherefore 
ABDC=BDEF=EFGH.  Q.  E.  D. 

Cor.  Hence  it  is  plain  that  triangles  on  the  same  or  equal 
bases  and  between  the  same  parallels  are  equal,  seeing  (by 
cor.  2,  theo.  12)  they  are  the  halves  of  their  respective  paral- 
lelogram. 

THEOREM  XIV. 
PL.  1.^.32. 

In  every  right-angled  triangle,  ABC,  the  square  of  the  hypothenuse  or 
longest  side,  BC,  or  BCMH,  is  equal  to  the  sum  of  the  squares  made  on  the 
other  two  sides  AB  and  AC,  that  is,  ABDE  and  ACGF. 

Through  A  draw  AKL  perpendicular  to  the  hypothenuse 
BC,  join  AH,  AM,  DC,  and  EG ;  in  the  triangles  BDC,  ABH, 
BD=BA,  being  sides  of  the  same  square,  and  also  BC=BH, 
and  the  included  angles  DBC=ABH  (for  DBA  =  CBH 
being  both  right,  to  both  add  ABC,  then  DBC=ABH),  there- 
fore the  triangle  DBC=ABH  (by  theo.  6) ;  but  the  triangle 
DBC  is  half  of  the  square  ABDE  (by  cor.  2,  theo.  12),  and 
the  triangle  ABH  is  half  the  parallelogram  BKLH.  The 
same  way  it  may  be  proved  that  the  square  ACGF  is  equal  to 
the  parallelogram  KCLM.  So  ABDE+ACGF  the  sum  of 
the  squares  =BKLH+KCML,  the  sum  of  the  two  parallelo- 
grams or  square  BCMH;  therefore  the  sum  of  the  squares  on 
AB  and  AC  is  equal  to  the  square  on  BC.  Q.  E.  D.* 

Cor.  1.  Hence  the  hypothenuse  of  a  right-angled  triangle 
may  be  found  by  having  the  sides  :  thus,  the  square  root  of  the 
sum  of  the  squares  of  the  base  and  perpendicular  will  be  the 
hypothenuse. 

*  For  different  demonstrations  of  this  excellent  theorem,  the  reader 
may  consult  Leslie's  Geometry  (Prop.  xi.  book  ii.). 


UNIVERSITY 

OF 


THEOREMS.  53 

Cor.  2.  Having  the  hypothenuse  and  one  side  given  to  find 
the  other ;  the  square  root  of  the  difference  of  the  squares  of 
the  hypothenuse  and  given  side  will  be  the  required  side. 

THEOREM  XV. 
PL.  1.^.33. 

In  all  circles  the  chord  of  60  degrees  is  always  equal  in  length  to  the 
radius. 

Thus  in  the  circle  AEBD,  if  the  arc  AEB  be  an  arc  of  60  degrees,  and 
the  chord  AB  be  drawn,  then  AB=CB=AC. 

In  the  triangle  ABC  the  angle  ACB  is  60  degrees,  being 
measured  by  the  arc  AEB ;  therefore  the  sum  of  the  other  two 
angles  is  120  degrees  (by  cor.  1,  theo.  5);  but  since  AC= 
CB,  the  angle  CAB=  CBA  (by  lemma  preceding  theo.  7); 
consequently  each  of  them  will  be  60,  the  half  of  120  degrees, 
and  the  three  angles  will  be  equal  to  one  another  as  well  as 
the  three  sides :  wherefore  AB=BC=AC.  Q.  E.  D. 

Cor.  Hence  the  radius  from  whence  the  lines  on  any 
scale  are  formed  is  the  chord  of  60  degrees  on  the  line  of 
chords. 

THEOREM  XVI. 
PL.  LJig.U. 

If  in  two  triangles,  ABC,  abc,  all  the  angles  of  one  be  each  respectively 
equal  to  all  the  angles  of  the  other ;  that  is,  A=a,  B=b,  C=c ;  then  the 
sides  opposite  to  the  equal  angles  will  be  proportional,  i)iz. 
AB:  ab:  :  AC  :  ac 
AB  :  ab  :  :  BC  :  be 
and  AC  :  ac  :  :  BC  :  be 

For  the  triangles  being  inscribed  in  two  circles,  it  is  plain, 
since  the  angle  A=a,  the  arc  BDC=*  bdc  and  consequently 
the  chord  BC  is  to  be  as  the  radius  of  the  circle  ABC  is  to 
the  radius  of  the  circle  abc  (for  the  greater  the  radius  is,  the 
greater  is  the  circle  described  by  that  radius  ;  and  consequently 
the  greater  any  particular  arc  of  that  circle  is,  so  the  chord,  sine, 
tangent,  &c.  of  that  arc  will  be  also  greater.  Therefore,  in 
general,  the  chord,  sine,  tangent,  <fcc.  of  any  arc  is  proportional 
to  the  radius  of  the  circle) ;  the  same  way  the  chord  AB  is  to 
the  chord  ab  in  the  same  proportion.  So  AB  :  ab  :  :  BC  : 
be.  The  same  way  the  rest  may  be  proved  to  be  propor- 
tional. 

*  The  arc  BDC  is  not  =  in  length  to  bdc,  as  might  be  supposed  from 
the  sign  of  equality  ;  but  they  contain  the  same  number  of  degrees,  as 
being  the  measure  of  equal  angles. 


54  GEOMETRICAL 


THEOREM  XVII. 
PL.  I.  Jig.  35. 

If  from  a  point  A  without  a  circle  DBCE  there  be  drawn  two  lines  ADE, 
ABC,  each  of  them  cutting  the  circle  in  two  points,  the  product  of  one 
whole  line  into  its  external  part,  viz.  AC  into  AB,  will  be  equal  to  that  of 
the  other  line  into  its  external  part,  viz.  AE  into  AD. 

Let  the  lines  DC,  BE  be  drawn  into  two  triangles  ABE, 
ADC-,  the  angle  AEB=ACD  (by  cor.  2,  theo.  7) ;  the  angle 
A  is  common,  and  (by  cor.  1,  theo.  5)  the  angle  ADC=ABE:> 
therefore  the  triangles  ABE,  AD  C  are  mutually  equiangular, 
and  consequently  (by  the  last)  AC  :  AE  :  :  AD  :  AB ',  where- 
fore AC  multiplied  by  AB  will  be  equal  to  AE  multiplied  by 
AD.  Q.  E.  D. 

THEOREM  XVIII. 

PL.  2.^-.  1. 

Triangles  ABC,  BCD,  and  parallelograms  ABCF  ana  BDEC,  having 
the  same  altitude,  have  the  same  proportion  between  themselves  as  their  bases 
BA  and  BD. 

Let  any  aliqu6t  part  of  AB  be  taken  which  will  also  measure 
BD :  suppose  that  to  be  ^g,  which  will  be  contained  twice  in 
AB,  and  three  times  in  BD,  the  parts  Ag,  gB,  Bh,  hi,  and  iD 
being  all  equal,  and  let  the  lines  gC,  AC,  and  iC  be  drawn: 
then  (by  cor.  to  theo.  13)  all  the  small  triangles  AgC,  gCB, 
BCh,  &c.  will  be  equal  to  each  other,  and  will  be  as  many  as 
the  parts  into  which  their  bases  were  divided  ;  therefore  it  will 
be,  as  the  sum  of  the  parts  in  one  base  is  to  the  sum  of  those 
in  the  other,  so  will  be  the  sum  of  the  small  triangles  in  the 
first  to  the  sum  of  the  small  triangles  in  the  second  triangle  ; 
that  is,  AB:  BD::  ABC  :  BDC. 

Whence  also  the  parallelograms  ABCF  and  BDEC,  being 
(by  cor.  2,  theo.  12)  the  doubles  of  the  triangles,  are  likewise 
as  their  bases.  Q.  E.  D. 

Note. — Wherever  there  are  several  quantities  connected  with 
the  sign  (:  :)  the  conclusion  is  always  drawn  fiom  the  first  two 
and  last  two  proportionals. 

THEOREM  XIX. 

PL.  2.  fig.Z. 

Triangles  ABC,  DEF,  standing  upon  equal  bases  AB  and  DE,  are  to 
each  other  as  their  altitudes  CG  and  FH. 

Let  BJ  be  perpendicular  to  AB  and  equal  to  CG,  in  which 
let  KB=FH,  and  let  AI  and  AK  be  drawn. 


THEOREMS.  55 


The  triangle  AIB=ACB  (by  cor.  to  theo.  13),  and 
DEF;  but  (by  theo.  18)  El :  BK  :  :  ABI :  ABK.     That  is, 
CG  :  FH :  :  ABC  :  DEF.     Q.  E.  D. 

THEOREM  XX. 
PL.  2.  Jig.  9. 

If  a  right  line  BE  be  drawn  parallel  to  one  side  of  a  triangle  ACD,  it 
will  cut  the  two  other  sides  proportionally,  viz.  AB  :  BC  :  :  AE  :  ED. 

Draw  CE  and  BD;  the  triangles  BEC  and  EBD  being  on 
the  same  base  BE  and  under  the  same  parallel  CD,  will  be 
equal  (by  cor.  to  theo.  13)  therefore  (by  theo.  18)  AB  :  BC  :  : 
(BEA  :  BEC  or  BEA  :  BED  :  :)  AE  :  ED.  Q.  E.  D. 

Cor.  1.  Hence  also  AC  :  AB :  :  AD  :  AE ;  for  AC  :  AB 
(AEC  :  AEB  :  :  ABD  :  AEB)  :  :  AD  :  AE. 

Cor.  2.  It  alsotftppears  that  a  right  line  which  divides  two 
sides  of  a  triangle  proportionally  must  be  parallel  to.  the  re- 
maining side. 

Cor.  3.  Hence,  also,  theo.  16  is  manifest;  since  the  sides 
of  the  triangles  ABE,  ACD,  being  equiangular,  are  propor- 
tional. 

THEOREM  XXI. 
PL.  2.  Jig.  4. 

If  two  triangles  ABC,  ADE  have  an  angle  BAG  in  the  one  equal  to 
an  angle  DAE  in  the  other,  and  the  sides  about  the  equal  angles  propor- 
tional ;  that  is,  AB  :  AD  : :  AC  :  AE ;  then  the  triangles  will  be  mutually 
equiangular. 

In  AB  take  Ad=AD,  and  let  de  be  parallel  to  BC,  meeting 
AC  inc. 

Because  (by  the  first  cor.  to  the  foregoing  theo.)  AB  :  Ad 
(or  AD)  :  :  AC  :  Ae,  and  (by  the  hypothesis,  or  what  is  given 
in  the  theorem)  AB  :  AD  :  :  AC:  AE ;  therefore  Ae=AE, 
seeing  AC  bears  the  same  proportion  to  each  ;  and  (by  theo. 
6)  the  triangle  Ade=ADE,  therefore  the  angle  Ade=D  and 
Aed=E;  but  since  ed  and  /JCare  parallel  (by  part  3,  theo.  3) 
Ade^B,  and  Aed=C,  therefore  B=D  and  C=E.  Q.  E.  D. 

THEOREM  XXII. 
PL.  2.  Jig.  5. 

Equiangular  triangles  ABC,  DEF  are  to  one  another  in  a  duplicate  pro- 
portion of  their  homologous  or  like  sides ;  or  as  the  squares  AK  and  DJtf 
of  their  homologous  sides.  ^ 

Let  the  perpendiculars  CG  and  FH  be  drawn,  as  well  as  the 
diagonals  BI  and  EL. 


56  GEOMETRICAL 

The  perpendiculars  make  the  triangles  ACG  and  DFH  equi- 
angular, and  therefore  similar  (by  theo.  16) ;  for  because  the 
angle  CAG=FDH,  and  the  right  angle  AGC=DHF>  the  re- 
maining angle  ACG—DFH  (by  cor.  2.  theo.  5). 

Therefore  GC  :  FH : :  (AC  :  DF : :)  AB :  DE,  or, which  is 
the  same  thing,  GC :  AB  :  :  FH  :  DE,  for  FH  multiplied  by 
AB=GC  multiplied  by  DE. 

By  theo.  19,  ABC:  ABI:  :(CG:  AI  or  AB  as  before:: 
FH:  DE,  or  DL):  :  DFE:  DLE,  therefore  ABC:  ABI: : 
DFE :  DLE,  or  ABC :  AK: :  DFE :  DM,  for  AK  is  double 
the  triangle  ABI>  and  DM  double  the  triangle  DEL,  (by  cor.  2. 
theo.  12.)  Q.  E.  D. 


THEOREM  XXIII. 
PL.  2. ./Eg-.  6. 

Like  polygons  ABODE,  abcde  are  in  a  duplicate  proportion  to  that  of 
the  sides  AB,  ab,  which  are  between  equal  angles  A  and  B  and  a  and  b,  or 
as  the  squares  of  the  sides  AB,  ab. 

Draw  AD,  AC,  ad,  ac. 

By  the  hypothesis  AB :  ab::BC:bc,  and  thereby  also  the 
angle  B—b-,  therefore  (by  theo.  21)  BAC=bac ;  and  ACS 
=acb  :  in  like  manner  EAD=ead,  and  EDA=eda.  If  there- 
fore from  the  equal  angles  A  and  a>  we  take  the  equal  ones 
EAD-}-BAC—ead-\-bac,  the  remaining  angle  DAC=dac, 
and  if  from  the  equal  angles  D  and  d,  EDA=eda  be  taken, 
we  shall  have  ADC=adc :  and  in  like  manner  if  from  C  and 
c  be  taken  BCA=bca,  we  shall  have  ACD=acd',  and  so 
the  respective  angles  in  every  triangle  will  be  equal  to  those 
in  the  other. 

By  theo.  22,  ABC:  abc  :  :  the  square  of  AC  to  the  square 
of  ac,  and  also  ADC :  adc: :  the  square  of  AC  to  the  square 
t)f  ae  ;  therefore,  from  equality  of  proportions,  ABC  :  abc  : : 
ADC  :  adc ;  in  like  manner  we  may  show  that  ADC  :  adc 
: :  EAD :  ead..  Therefore  it  will  be,  as  one  antecedent  is  to  one 
consequent,  so  are  all  the  antecedents  to  all  the  consequents. 
That  is,  ABC  is  to  abc  as  the  sum  of  the  three  triangles  in 
the  first  polygon  is  to  the  sum  of  those  in  the  last.  Or  ABC 
will  be  to  abc  as  polygon  to  polygon. 

The  proportion  of  ABC  to  abc  (by  the  foregoing  theo.)  is 
Us  the  square  of  AB  is  to  the  square  of  ab,  but  the  proportion 
of  polygon  to  polygon  is  as  ABC  to  abc,  as  now  shown: 
therefore  the  proportion  of  polygon  to  polygon  is  as  the  square 
of  AB  to  the  square  of  ab. 


THEOREMS.  57 

THEOREM  XXIV. 
PL.  I.  Jig.  8. 

Let  DHB  be  a  quadrant  of  a  circle  described  by  the  radius  CB ;  HB  an 
arc  of  it,  and  DH  its  complement ;  HL  or  FC  the  sine,  FHor  CL  its  co- 
sine, BK  its  tangent,  DI  its  cotangent ;  CK  its  secant,  and  CI  its  co- 
secant.. Fig.  8. 

1.  The  cosine  of  an  arc  is  to  the  sine  as  the  radius  is  to  the 
tangent. 

2.  The  radius  is  to  the  tangent  of  an  arc  as  the  cosine  of  it 
is  to  the  sine. 

3.  The  sine  of  an  arc  is  to  its  cosine  as  the  radius  to  its 
cotangent. 

4.  Or  the  radius  is  to  the  cotangent  of  an  arc  as  its  sine  to  its 
co-sine. 

5.  The  cotangent  of  an  arc  is  to  the  radius  as  the  radius  to 
the  tangent. 

6.  The  cosine  of  an  arc  is  to  the  radius  as  the  radius  is  to 
the  secant. 

7.  The  sine  of  an  arc  is  to  the  radius  as  the  tangent  is  to 
the  secant. 

The  triangles  CLH  and  CBK  being  similar  (by  theo.  16), 

1.  CL:LH::  CB  :  BK. 

2.  Or,  CB  :  BK  :  :  CL  :  LH. 

The  triangles  CFH  and  CDI  being  similar, 

3.  CF  (or  LH )  :  FH  :  :  CD  :  DI. 

4.  CD:  DI::  CF  (or  LH ) :  FH. 

The  triangles  CDI  and  CBK  are  similar;  for  the  angle 
CID=KCB,  being  alternate  ones  (by  part  2,  theo.  3),  the 
lines  CB  and  DI  being  parallel,  the  angle  CDI=CBK 
being  both  right,  and  consequently  the  angle  DCI=CKB> 
wherefore, 

5.  DI :  CD  : :  CB  :  BK. 

And  again,  making  use  of  the  similar  triangle  CLH  and 
CBK, 

6.  CL  :  CB  :  :  CH :  CK. 

7.  HLiCH-.iBK:  CK. 


58  GEOMETRICAL 


GEOMETRICAL  PROBLEMS. 

•.  0.  "»  if-vVj     •    .-,\r  ;  ')  Wsna^w  iV\r>  -T<  "i>  '•'      ;-^v;.  . 


PROBLEM  I. 
PL.  2.^.7. 

To  wta&e  a  triangle  of  three  given  right  lines  B  0,  LB,  L  0,  of  which  any 
two  must  be  greater  than  the  third. 

Lay  BL  from  B  to  L',  from  B  with  the  line  BO  describe 
an  are,  and  from  L  with  LO  describe  another  arc  ;  from  O,  the 
intersecting  point  of  those  arcs,  draw  BO  and  OL,  and  BOL 
ijS  the  triangle  required. 

This  is  manifest  from  the  construction. 


PROBLEM  II. 

PL.  2.  Jig.  8. 

At  a  point  B  in  a  given  right  line  BC,  to  make  an  angle  equal  to  a  given 
angle  A. 

Draw  any  right  line  ED  to  form  a  triangle,  as  EAD,  take  BF 
=ADj  and  upon  /£Fmake  the  triangle  BFG,  whose  side  BG= 
AE,  and  GF=ED  (by  the  last),,  then  also  the  angle  B= A  ; 
if  we  suppose  one  triangle  be  laid'  on  the  other,  the  sides  will 
mutually  agree  with'  each  other,  and  therefore  be  equal ;  for 
if  we  consider  these  two  triangles. to  be  made  of  the  same 
three  given  lines,  they  are  manifestly  one  and  the  same  triangle. 

Otherwise, 

Upon  the  centres  A  and  B,  at  any  distance,  let  two  arcs 
DE,  FG,  be  described;  make  the  arc  FG=DE,  and  through 
B  and  6r  draw  the  line  BG,  and  it  is  done. 

For  since  the  chords  ED,  GF  are  equal,  the  angles  A  and 
B  are  also  equal,  as  before  (by  def.  17); 

PROBLEM  £L 

PL.  2.  Jig.  9. 

To  bisect  or  divide  into  two  'equal  parts  any  given  right-lined  angle  BA  C. 
In  the  lines  AB  and  AC,  from  the  point  A,  set  off  equal  dis- 
tances AE=AD',  then,  with  any  distance  more  than  the  half 
of  DE,  describe  two  arcs  to  cut  each  other  in  some  point  JF; 
and  the  right  line  AF,  joining  the  points  A  and  F,  will  bisect 
the  given  angle  BAG. 

For  if  ZXFand  FE  be  drawn,  the  triangles  ADF,  AEF  are 
equilateral  to  each  other,  viz.  AD= AE,  DF=FE,  and  AF 
common,  wherefore  DAF=EAF,  as  before. 


PROBLEMS.  59 

PROBLEM  IV. 

PL.  2.  Jig.  10. 

To  bisect  a  right  line  AB. 

With  any  distance  more  than  half  the  line  from  A  and  B, 
describe  two  circles  CFD,  CGD,  cutting  each  other  in 
the  points  C  and  D;  draw  CD  intersecting  AB  in  E,  then 


For,  if  AC,  AD,  BC,  BD  be  drawn,  the  triangles  ACD, 
BCD  will  be  mutually  equilateral,  and  consequently  the  angle 
ACE=BCE;  therefore  the  triangle  ACE,  BCE,  having 
AC=BC,  CE  common,  and  the  angle  ACE=BCE  ;  (by  theo. 
6)  the  base  AE=the  base  BE. 

Cor.  Hence  it  is  manifest  that  CD  not  only  bisects  AB,  but 
is  perpendicular  to  it  (by  def.  10). 

PROBLEM  V. 

PL.  2.  fig.  II. 

On  a  given  point  A,  in  a  right  line  EF,  to  erect  a  perpendicular. 
From  the  point  A  lay  off  on  each  side  the  equal  distances 
AC,  AD;  and  from  C  and  D  as  centres,  with  any  interval 
greater  than  AC  or  AD,  describe  two  arcs  intersecting  each 
other  in  n  ;  from  A  to  B  draw  the  line  AB,  and  it  will  be  the 
perpendicular  required. 

For  let  CB  and  BD  be  drawn,  then  the  triangles  CAB,  DAB 
will  be  mutually  equilateral  and  equiangular,  so  CAB=DAB9 
a  right  angle  (by  def.  10). 

PROBLEM  VI. 

PL.  2.  fig.  12. 

To  raise  a  perpendicular  on  the  end  B  of  a  right  line  AB. 
From  any  point  D  not  in  the  line  AB,  with  the  distance  from 
D  to  B,  let  a  circle  be  described  cutting  AB  in  E  ;  draw  from 
E  through  D  the  right  line  EDC,  cutting  the  periphery  in  C, 
and  join  CB,  and  that  is  the  perpendicular  required. 

EBC  being  a  semicircle,  the  angle  EBC  will  be  a  right 
angle  (by  cor.  5,  theo.  7). 

ROBLEM  VII. 
PL.  2.  fig.  13. 

From  a  given  point  At  to  let  fall  a  perpendicular  upon  a  given  right 
line  BC. 

From  any  point  D,  in  the  given  line,  take  the  distance  to  th« 
given  point  A,  and  with  it  describe  a  circle  AGE,  make  GJB= 


00  GEOMETRICAL 

AG,  join  the  points  A  and  E  by  the  line  AFEf  and  AF  will 
be  the  perpendicular  required. 

Let  DA,  DE  be  drawn,  the  angle  ADF—FDE,  DA=DEt 
being  radii  of  the  same  circle,  and  DF  common;  there- 
fore (by  theo.  6)  the  angle  DFA=DFE,  and  FA  a  perpen- 
dicular. (By  defl  10.) 

PROBLEM  VIIL 
PL.  2.  fig.  14. 

Through  a  given  point  A  to  draw  a  right  line  AB,  'parallel  to  a  given 
right  line  CD. 

•  From  the  point  A  to  any  point  F  in  the  line  CD  draw  the 
line  AF',  with  the  interval  FA,  and  onefoot  of  the  compasses 
in  F,  describe  the  arc  AE,  and  with  the  like  interval  and  one 
foot  in  A  describe  the  arc  BF,  making  BF—AE',  through  A 
and  B  draw  the  line  AB,  and  it  will  be  parallel  to  CD. 

By  prob.  2,  The  angle  BAF=AFE,  and  by  theo.  11,  BA 
and  CD  are  parallel. 

PROBLEM  IX. 

PL.  I.  Jig.  17. 
Upon  a  given  line  AB  to  describe  a  square  ABCD* 

Make  BC  perpendicular  and  equal  to  AB,  and  from  A  and 
<7,  with  the  line  AB  or  BC,  let  two  arcs  be  described,  cutting 
each  other  in  D ;  from  whence  to  A  and  C  let  the  lines  AD, 
DC  be  drawn  ;  so  is  ABCD  the  square  required. 

For  all  the  sides  are  equal  by  construction  ;  therefore  the 
triangles  ADC  and  BAC  are  mutually  equilateral  and  equian- 
gular, and  ABCD  is  an  equilateral  parallelogram,  whose  angles 
are  right.  For  B  being  right,  D  is  also  right,  and  DAC, 
DC  A,  BAC,  ACS,  each  half  a  right  angle  (by  lemma  preced- 
ing theo,  7,  and  cor.  2,  theo.  5),  whence  DAB  and  BCD 
will  each  be  a  right  angle,  and  (by  def.  43)  ABCD  is  a 
square. 

SCHOLIUM, 

By  the  same  method  a  rectangle  or  oblong  may  be  described, 
the  sides  thereof  being  given. 

PROBLEM  X. 
PL.  2.  Jiff.  15. 

To  divide  a  given  right  line  AB  into  any  proposed  number  of  equal 
parts. 

Draw  the  indefinite  right  line  AP,  making  any  angle  with 


PROBLEMS,  61 

AB,  also  draw  BQ  parallel  to  AP,  in  each  of  which  let  there 
be  taken  as  many  equal  parts  AM,  MN,  &c.  Bo,  on,  &c.  as 
you  would  have  AB  divided  into  ;  then  draw  Mm,  JVn,  &c» 
intersecting  AB  in  £,  F,  &c.  and  it  is  done. 

For  MN  and  mn  being  equal  and  parallel,  FN  will  be  par- 
allel to  EM,  and  in  the  same  manner  GO  to  FN(by  theo.  12)  ; 
therefore  AM,  MN,  NO,  being  all  equal  by  construction,  it  is 
plain  (from  theo.  20)  that  AEf  EF9  FG,  &c.  will  likewise  be 
equal. 

PROBLEM  XL 

PL.  &  fig.  16. 

To  find  a  third  proportional  to  two  given  right  lines  A  and  B. 
Draw  two  indefinite  blank  lines  CE,  CD  anywise  to  make 
any  angle.  Lay  the  line  A  from  C  to  F,  and  the  line  B  from 
C  to  G,  and  draw  the  line  FG  ;  lay  again  the  line  A  from  C 
to  H,  and  through  #draw  HI  parallel  to  FG  (by  prob.  8),  so 
is  CI  the  third  proportional  required. 

For,  by  cor.  1,  theo.  20,  CG  :  CH  :  :  CF  :  CL 


PROBLEM  XII. 

PL.  2.  fig.  17. 

Three  right  lines  A,  B,  C  given,  to  find  a  fourth  proportional. 
Having  made  an  angle  DEF  anywise,  by  two  indefinite 
blank  right  lines  ED,  EF,  as  before  ;  lay  the  line  A  from  E 
to  fr,  the  line  B  from  E  to  /,  and  draw  the  line  IG  ;  lay  the 
line  C  from  E  to  H,  and  (by  prob.  8)  draw  HK  parallel  there- 
to, so  will  EK  be  the  fourth  proportional  required. 
For,  by  cor.  1,  theo.  20,  £6?  :  El  :  :  EH  :  EK. 
Ox,AiB::C:EK. 

PROBLEM  XIII. 

PL.  3.  fig.  1. 

Two  right  lines  A  and  B  given,  to  find  a  mean  proportional. 
Draw  an  indefinite  straight  line,  on  which  place  AB=A 
and  BC=B  ;  bisect  AC  (by  prob.  4)  in  E,  and  describe  the 
semicircle  ADC,  and  from  the  point  B  erect  the  perpendicular 
BD  (by  prob.  5),  then  BD  is  a  mean  proportional. 

For  if  the  lines  AD,  DC  be  drawn,  the  angle  ADC  is  a 
right  angle  (by  cox.  5*  theo.  7),  being  an  angle  in  a  semi- 
circle. 

The  angles  ABD,  DBC  are  right  ones  (by  def.  10),  the  line 
BD  being  a  perpendicular  ;  wherefore  the  triangles  ABD, 


62  GEOMETRICAL 

DBG  are  similar  :  thus  the  angle  ABD=DBC,  being  both 
right,  the  angle  DAC  is  the  complement  of  BDA  to  a  right 
angle  (by  cor.  2,  theo.  5),  and  is  therefore  equal  to  BDC,  the 
angle  ADC  being  a  right  angle  as  before  ;  consequently  (by 
cor.  1,  theo.  5)  the  angle  ADB=DCB;  wherefore  (by 
theo.  16), 

AB  :  BD  ::  BD:  BC 

Of,  A  :  BD  ::BD:  B. 

PROBLEM  XIV. 
PL.  3.^.2. 

To  divide  a  right  line  AB  in  the  point  E,  so  that  AE  shall  have  the  same 
proportion  to  EB  as  two  given  lines  C  and  D  have. 

Draw  an  indefinite  blank  line  AF  to  the  extremity  of  the 
line  AB,  to  make  with  it  "any  angle  ;  lay  the  line  C.from  A  to 
C,  and  D  from  C  to  D,  and  join  the  points  B  and  D  by  the 
line  BD  ;  through  C  draw  CE  parallel  to  BD  (by  prob.  8),  so 
is  E  the  point  of  division. 

For,  by  theo.  20,  AC  :  CD  :  :  AE  :  EB. 

Or,  C  :  D  :  :  AE  :  :  EB. 

PROBLEM  XV. 
PL.  3.  fig.  3. 

To  describe  a  circle  about  a  triangle  ABC,  or  (which  is  the  same  thing) 
through  any  three  points  A,  B,  C,  which  are  not  situated,  in  a  right  line. 

By  prob.  4.  Bisect  the  line  A  C  by  the  perpendicular  DE, 
and  also  CB  by  the  perpendicular  FG,  the  point  of  intersection 
H  of  these  perpendiculars  is  the  centre  of  the  circle  required; 
from  which  take  the  distance  to  any  of  the  three  points  A,  B, 
C,  and  describe  the  circle  ABC,  and  it  is  done. 

For,  by  cor.  ^to  theo.  8,  the  lines  DE  and  FGr  must  each 
pass  through  the  centre  ;  therefore  their  point  of  intersection 
H  must  be  the  centre. 

SCHOLIUM. 

By  this  method  the  centre  of  a  circle  may  be  found,  by  hav- 
ing only  a  segment  of  it  given. 


PROBLEM  XYL 
PL.  3.  fig.  4. 

To  make  an  angle  of  any  number  of  degrees  at  the  point  A  of  the  line  AB, 
suppose  of  45  degrees.' 

From  a  scale  of  chords  take  60  degrees,  for  60°  is  equal  to 
the  radius  (by  cor.  theo.  15),  and  with  that  distance  from  A  as 
a  centre,  describe  a  circle  from  the  line  A  B  ;  take  45  degrees, 


PROBLEMS.  63 

the  quantity  of  the  given  angle,  from  the  same  scale  of  chords, 
and  lay  it  on  that  circle  from  a  to  b ;  through  A  and  b  draw 
the  line  AbC,  and  the  angle  A  will  be  an  angle  of  45  degrees,, 
as  required. 

If  the  given  angle  be  more  than  90°,  take  its  half  (or  divide 
it  into  any  two  parts  less  than  90  and  lay  them  after  each  other 
on  the  arc,  which  is  described  with  the  chord  of  60  degrees  ; 
through  the  extremity  of  which  and  the  centre,  let  a  Ine  be 
drawn,  and  that  will  form  the  angle  required,  with  the  given 
line. 

PROBLEM  XVII. 

PL.  3.  fig.  5. 

To  measure  a  given  angle  ABC. 

If  the  lines  which  include  the  angle  be  not  as  long  as  the 
chord  of  60°  on  your  scale,  produce  them  to  that  or  a  greater 
length,  and  between  them  so  produced,  with  the  chord  of  60° 
from  J5,  describe  the  arc  ed ;  which  distance  ed,  measured  on 
the  same  line  of  chords,  gives  the  quantity  of  the  angle  ABC, 
as  required ;  this  is  plain  from  def.  17. 

PROBLEM  XVIII. 
PL.  3.  fig.  6. 

To  make  a  triangle  BCE  equal  to  a  given  quadrilateral  figure  ABCD. 

Draw  the  diagonal  AC,  and  parallel  to  it  (by  prob.  8)  DEt 
meeting  AB  produced  in  E ;  then  draw  CE,  and  ECB  will 
be  the  triangle  required. 

For  the  triangles  ADC,  AEC  being  upon  the  same  base 
AC,  and  under  the  same  parallel  ED  (by  eor.  to  theo.  13), 
will  be  equal,  therefore  if  ABC  be  added  to  each,  then  ABCD 
=BEC. 

PROBLEM  XIX. 
PL.  3,  fig.  It. 

To  make  a  triangle  DFH  equal  to  a  given  five-sided  figure  ABODE. 

Draw  DA  and  DB,  and  also  EH  and  C F  parallel  to  them, 
(by  prob.  8)  meeting  AB  produced  in  H  and  F;  then  draw 
DH,  DF,  and  the  triangle  HDF  is  the  one  required. 

For  the  triangle  DEA=DHA,  and  DBC=DFB  (by  cor. 
to  theo.  13)  ;  therefore  by  adding  these  equations,  DEA+DBC 
=DHA+DFB,if  to  each  of  these  ADB  be  added;  then 
DEA+ADB+DBC=ABCDE  (^ 
DHF. 


64  MATHEMATICAL 

PROBLEM  XX. 

PL.  3.  jig.  8. 

To  project  the  lines  of  chords,  sines,  tangents,  and  secants  vrith  any  ra&iuit. 
On  the  line  AB,  let  a  semicircle  ADB  be  described ;  let 
CDF  be  draWn  perpendicular  to  this  line  from  the  centre  C ; 
and  the  tangent  BE  perpendicular  to  the  end  of  the  diameter  ; 
let  the  quadrants  AD,  DB  be  each  divided  into  nine  equal 
parts,  ^very  one  of  which  will  be  ten  degrees;  if  then  from  the 
centre  C  lines  be  drawn  through  10,  20,  30,  40,  &c.  the  divi- 
sions of  the  quadrant  BD,  and  continued  to  BE,  we  shall  there 
have  the  tangents  of  10,  20,  30,  40,  &c.  and  the  secants  C 
10,  C  20,  C  30,  &c.  are  transferred  to  the  line  CF,  by  describing 
the  arcs  10, 10 ;  20,  20 ;  30,  30,  <fcc.  If  from  10,  20,  30,  &c. 
the  divisions  of  the  quadrant  BD,  there  be  let  fall  perpendicu- 
lars, let  these  be  transferred  to  the  radius  CB,  and  we  shall 
have  the  sines  of  10,  20  30,  &c.  and  if  from  A  we  describe 
the  arcs  10,  10 ;  20,  20 ;  30,  30,  &c.  from  every  division  of 
the  arc  AD,  we  shall  have  a  line  of  chords.  The  same  way 
we  may  have  the  sine,  tangent,  &c.  to  every  single  degree  on 
the  quadrant,  by  subdividing  each  of  the  nine  former  divisions 
into  ten  equal  parts.  By  this  method  the  sines,  tangents,  &c. 
may  be  drawn  to  any  radius ;  and  then,  after  they  are  trans- 
ferred to  lines  on  a  rule,  we  shall  have  the  scales  of  sines, 
tangents,  &c.  ready  for  use. 


MATHEMATICAL 
DRAWING  INSTRUMENTS. 

THE  strictness  of  geometrical  demonstration  admits  of  no 
other  instruments  than  a  rule  and  a  pair  of  compasses.  .  But, 
in  proportion  as  the  practice  of  geometry  was  extended  to  the 
different  arts,  either  connected  with  or  dependent  upon  it, 
new  instruments  became  necessary,  some  to  answer  peculiar 
purposes,  some  to  facilitate  operation,  and  others  to  promote 
accuracy. 

As  almost  every  artist  whose  operations  are  connected  with 
mathematical  designing  furnishes  himself  with  a  case  of 
drawing  instruments  suited  to  his  peculiar  purposes,  they  are 
fitted  up  in  various  modes,  some  containing  more,  others  fewer 
instruments.  The  smallest  collection  put  into  a  case  consists 
of  a  plane  scale,  a  pair  of  compasses  with  a  moveable  leg,  and 
two  spare  points,  which  may  be  applied  occasionally  to  the 


DRAWING  INSTRUMENTS.  65 

compasses ;  one  of  these  points  is  to  hold  ink ;  the  other  a 
portcrayon,  for  holding  a  piece  of  black-lead  pencil. 

What  is  called  a  full  pocket  case,  contains  the  following  in 
struments. 

A  pair  of  large  compasses  with  a  moveable  point,  an  ink 
point,  a  pencil  point,  and  one  for  dotting ;  either  of  those  points 
may  be  inserted  in  the  compasses  instead  of  the  moveable  leg. 

A  pair  of  plain  compasses  somewhat  smaller  than  those 
with  the  moveable  leg. 

A  pair  of  bow  compasses. 

A  drawing  pen  with  a  protracting  pin  in  the  upper  part. 

A  sector. 

A  plain  scale. 

A  protractor. 

A  parallel  rule. 

A  pencil  and  screwdriver.* 

*  Large  collections  are  called  magazine  cases  of  instruments ;  these 
generally  contain — 

A  pair  of  six  inch  compasses  with  a  moveable  leg,  an  ink  point,  a 
dotting  point,  the  crayon  point,  so  contrived  as  to  hold  a  whole  pencil,  two 
additional  pieces  to  lengthen  occasionally  one  leg  of  the  compasses,  and 
thereby  enable  them  to  measure  greater  extents,  and  describe  circles  of  a 
larger  radius. 

A  pair  of  hair  compasses. 

A  pair  of  bow  compasses. 

A  pair  of  triangular  compasses. 

A  sector. 

A  parallel  rule. 

A  protractor. 

A  pair  of  proportional  compasses,  either  whh  or  without  an  adjusting 
screw. 

A  pah*  of  wholes  and  halves. 

Two  drawing  pens,  and  a  pointril. 

A  pair  of  small  hair  compasses,  with  a  head  similar  to  those  of  the  bow 
compasses. 

A  knife,  a  file,  a  key,  and  screwdriver,  or  the  compasses  in  one  piece. 

A  small  set  of  fine  water-colours. 

To  these  some  of  the  following  instruments  are  often  added : — 

A  pair  of  beam  compasses. 

A  pair  of  gunners'  callipers. 

A  pair  of  elliptical  compasses. 

A  pair  of  spiral  compasses. 

A  pair  of  perspective  compasses. 

A  pair  of  compasses  with  a  micrometer  screw. 

A  rule  for  drawing  lines,  tending  to  a  centre  at  a  great  distance. 

A  protractor  and  parallel  rule. 

One  or  more  parallel  rules. 

A  pantographer,  or  pentagraph. 

A  pair  of  sectoral  compasses,  forming  at  the  same  time  a  pair  of  beam 
and  calliper  compasses. 


66  MATHEMATICAL 

In  a  case  with  the  best  instruments,  the  protractor  and  plain 
scale  are  always  combined.  The  instruments  inmost  general 
use  are  those  of  six  inches ;  instruments  are  seldom  made 
longer,  but  often  smaller.  Those  of  six  inches  are,  however, 
to  be  preferred,  in  general,  before  any  other  size ;  they  will 
effect  all  that  can  be  performed  with  the  shortest  ones,  while, 
at  the  same  time,  they  are  better  adapted  to  large  works. 

OF    DRAWING    COMPASSES. 

Compasses  are  made  -either  of  silver  or  brass,  but  with  steel 
points.  The  joints  should  always  be  framed  of  different  sub- 
stances ;  thus,  one  side  or  part  should  be  of  silver  or  brass, 
and  the  other  of  steel.  The  difference  in  the  texture  and  pores 
of  the  two  metals  causes  the  parts  to  adhere  less  together,  di- 
minishes the  wear,  and  promotes  uniformity  in  their  motion. 
The  truth  of  the  work  is  ascertained  by  the  smoothness  and 
equality  of  the  motion  at  the  joint,  for  all  shake  and  irregularity 
is  a  certain *sign  of  imperfection.  The  points  should  be  of 
steel,  so  tempered  as  neither  to  be  easily  bent  or  blunted ;  not 
too  fine  and  tapering,  and  yet  meeting  closely  when  the  com- 
passes are  shut. 

As  an  instrument  of  art,  compasses  are  so  well  known  that 
it  would  be  superfluous  to  enumerate  their  various  uses ;  suffice 
it  then  to  say,  that  they  are  used  to  transfer  small  distances, 
measure  given  spaces,  and  describe  arches  and  circles. 

If  an  arch  or  circle  is  to  be  described  obscurely,  the  steel 
points  are  best  adapted  to  the  purpose  ;  if  it  is  to  be  in  ink  or 
black  lead,  either  the  drawing  pen,  or  crayon  points  are  to  be 
used. 

To  use  a  pair  of  compasses.  Place  the  thumb  and  middle 
finger  of  the  right  hand  in  the  opposite  hollowsnn  the  shanks 
of  the  compasses,  then  press  the  compasses,  and  the  legs  will 
open  a  little  way ;  this  being  done,  push  the  innermost  leg 
with  the  third  finger,  elevating  at  the  same  time  the  furthermost 
with  the  nail  of  the  middle  finger,  till  the  compasses  are  suffi- 
ciently opened  to  receive  the  middle  and  third  finger ;  they  may 
then  be  extended  at  pleasure,  by  pushing  the  furthermost  leg 
outwards  with  the  middle,  or  pressing  it  inwards  with  the  fore- 
finger. In  describing  circles  or  arches,  set  one  foot  of  the 
compasses  on  the  centre,  and  then  roll  the  head  of  the  com- 
passes between  the  middle  and  fore-finger,  the  other  point 
pressing  at  the  same  time  upon  the  paper.  They  should  be 
held  as  upright  as  possible,  and  care  should  be  taken  not  to 
press  forcibly  upon  them,  but  rather  to  let  them  act  by  their 


DRAWING  INSTRUMENTS.  67 

own  weight ;  the  legs  should  never  be  so  far  extended  as  to 
form  an  obtuse  angle  with  the  paper  or  plane  on  which  they 
are  used. 

The  ink  and  crayon  points  have  a  joint  just  under  that  part 
which  fits  into  the  compasses  ;  by  this  they  may  be  always  so 
placed  as  to  be  set  nearly  perpendicular  to  the  paper ;  the  end 
of  the  shank  of  the  best  compasses  is  framed  so  as  to  form  a 
strong  spring,  to  bind  firmly  the  moveable  points,  and  prevent 
them  from  shaking.  This  is  found  to  be  a  more  effectual 
method  than  that  by  a  screw. 

Two  additional  pieces  are  often  applied  to  these  compasses ; 
these,  by  lengthening  the  leg,  enable  them  to  strike  larger  cir- 
cles, or  measure  greater  extents,  than  they  would  otherwise 
perform,  and  that  without  the  inconveniences  attending  longer 
compasses.  When  compasses  are  furnished  with  this  additional 
piece,  the  moveable  leg  has  a  joint,  that  it  may  be  placed  per- 
pendicular to  the  paper. 

The  bow  compasses  are  a  small  pair,  usually  with  a  point 
for  ink ;  they  are  used  to  describe  small  arches  or  circles, 
which  they  do  much  more  conveniently  than  large  com- 
passes, not  only  on  account  of  their  size,  but  also  from  the 
shape  of  the  head,  which  rolls  with  great  ease  between  the 
fingers. 

Of  the  drawing  pen  and  protracting  pin.  The  pen  part  of 
this  instrument  is  used  to  draw  straight  lines :  it  consists  of 
two  blades  with  steel  points  fixed  to  a  handle  ;  the  blades  are 
so  bent  that  the  ends  of  the  steel  points  meet,  and  yet  leave  a 
sufficient  cavity  for  the  ink ;  the  blades  may  be  opened  more 
or  less  by  a  screw,  and,  being  properly  set,  will  draw  a  line  of 
any  assigned  thickness.  One  of  the  blades  is  framed  with  a 
joint,  that  the  points  may  be  separated  and  thus  cleaned  more 
conveniently.  A  small  quantity  only  of  ink  should  be  put  at 
one  time  into  the  drawing  pen,  and  this  should  be  placed  in  the 
cavity  between  the  blades  by  a  common  pen  or  feeder ;  the 
drawing  pen  acts  better  if  the  pen  by  which  the  ink  is  inserted 
be  made  to  pass  through  the  blades.  To  use  the  drawing  pen, 
first  feed  it  with  ink,  then  regulate  it  to  the  thickness  of  the 
required  line  by  the  screw.  In  drawing  lines,  incline  the  pen 
a  small  degree,  taking  care,  however,  that  the  edges  of  both  the 
blades  touch  the  paper,  keeping  the  pen  close  to  the  rule,  and 
in  the  same  direction  during  the  whole  operation.  The 
blades  should  always  be  wiped  very  clean  before  the  penis  put 
away. 

These  directions  are  equally  applicable  to  the  ink  point  of 
the  compasses,  only  observing,  that  when  an  arch  or  circle 


68  MATHEMATICAL 

is  to  be  described  of  more  than  an  inch  radius,  the  point 
should  be  so  bent  that  the  blades  of  the  pen  may  be  nearly 
perpendicular  to  the  paper,  and  both  of  them  touch  it  at  the 
same  time. 

,  The  protracting  pin  is  only  a  short  piece  of  steel  wire  with 
a  very  fine  point  fixed  at  one  end  of  the  upper  part  of  the  han- 
dle of  the  drawing  pen.  It  is  used  to  mark  the  intersection 
of  lines,  or  to  set  off  divisions  from  the  plotting  scale  and  pro- 
tractor. 

OF    THE    SECTOR. 

Amid  the  variety  of  mathematical  instruments  that  have 
been  contrived  to  facilitate  the  art  of  drawing,  there  is  none  so 
extensive  in  its  use  or  of  such  general  application  as  the  sector. 
It  is  a  universal  scale,  uniting,  as  it  were,  angles  and  parallel 
lines,  the  rule  and  the  compass,  which  are  the  only  means  that 
geometry  makes  use  of  for  measuring,  whether  in  speculation 
or  practice.  The  real  inventor  of  this  valuable  instrument  is 
unknown ;  yet  of  so  much  merit  has  the  invention  appeared, 
that  it  was  claimed  by  Galileo,  and  disputed  by  nations. 

This  instrument  derives  its  name  from  the  tenth  definition 
of  the  third  book  of  Euclid,  where  he  defines  the  sector  of 
a  circle.  It  is  formed  of  two  equal  rules  called  legs ;  these 
legs  are  moveable  about  the  centre  of  a  joint,  and  will,  conse- 
quently, by  their  different  openings,  represent  every  possible 
variety  of  plane  angles.  The  distance  of  the  extremities  of 
these  rules  are  the  subtenses  or  chords,  or  the  arches  they 
describe. 

Sectors  are  made  of  different  sizes,  but  their  length  is  usually 
denominated  from  the  length  of  the  legs  when  the  sector  is  shut. 
Thus  a  sector  of  six  inches  when  the  legs  are  close  together 
forms  a  rule  of  twelve  inches  when  opened ;  and  a  foot  sector 
is  two  feet  long  when  opened  to  its  greatest  extent.  In  de- 
scribing the  lines  usually  placed  on  this  instrument,  I  refer  to 
those  commonly  laid  down  on  the  best  six-inch  brass  sectors. 
But  as  the  principles  are  the  same  in  all,  and  the  differences 
little  more  than  in  the  number  of  subdivisions,  it  is  to  be  pre- 
sumed that  no  difficulty  will  occur  in  the  application  of  what 
is  here  said  to  sectors  of  a  larger  radius. 

The  scales,  or  lines  graduated  upon  the  faces  of  the  instru- 
ment, and  which  are  to  be  used  as  sectoral  lines,  proceed  from 
the  centre,  and  are,  1.  Two  scales  of  equal  parts,  one  on  each 
leg,  marked  LIN.  or  L.  Each  of  these  scales,  from  the  great 
extensiveness  of  its  use,  is  called  the  line  of  lines.  2.  Two 
lines  of  chords,  marked  CHO.  or  c.  3.  Two  lines  of  secants* 


DRAWING  INSTRUMENTS.  69 

marked  SEC.  or  s.  A  line  of  polygons,  marked  POL.  Upon  the 
other  face  the  sectoral  lines  are,  1.  Two  lines  of  sines  marked 
SIN.  or  s.  2.  Two  lines  of  tangents,  marked  TAN.  3.  Be- 
tween the  lines  of  tangents  and  sines  there  is  another  line  of 
tangents  to  a  lesser  radius,  to  supply  the  defect  of  the  former, 
and  extending  from  45°  to  75°. 

Each  pair  of  these  lines,  except  the  line  of  polygons,  is  so 
adjusted  as  to  make  equal  angles  at  the  centre,  and  conse- 
quently at  whatever,  distance  the  sector  be  opened,  the  angles 
will  be  always  respectively  equal.  That  is,  the  distance  be- 
tween 10  and  10  on  the  line  of  lines  will  be  equal  to  60  and 
60  on  the  line  of  chords,  90  and  90  on  the  line  of  sines,  and  45 
and  45  on  the  line  of  tangents. 

Besides  the  sectoral  scales^  there  are  others  on  each  face 
placed  parallel  to  the  outward  edges,  and  used  as  those  of  the 
common  plain  scale.  There  are  on  the  one  face,  1.  A  line  of 
inches.  2.  A  line  of  latitudes.  3.  A  line  of  hours.  4.  A 
line  of  inclination  of  meridians.  5.  A  line  of  chords.  On  the 
other  face,  three  logarithmic  scales,  namely,  one  of  numbers, 
one  of  sines,  and  one  of  tangents  :  these  are  used  when  the 
sector  is  fully  opened,  the  legs  forming  one  line. 

To  read  and  estimate  the  divisions  on  the  sectoral  lines.  The 
value  of  the  divisions  on  most  of  the  lines  is  determined  by  the 
figures  adjacent  to  them ;  these  proceed  by  tens,  which  consti- 
tute the  divisions  of  the  first  order,  and  are  numbered  accord- 
ingly ;  but  the  value  of  the  divisions  on  the  line  of  lines,  that 
are  distinguished  by  figures,  is  entirely  arbitrary,  and  may 
represent  any  value  that  is  given  to  them ;  hence  the  figures 
1,  2,  3,  4,  &c.  may  denote  either  10,  20,  30,  40,  or  100,  200, 
300,  400,  and  so  on. 

The  line  of  lines  is  divided  into  ten  equal  parts,  numbered 
1,  2,  3,  to  10  ;  these  may  be  called  divisions  of  the  first  order ; 
each  of  these  is  again  subdivided  into  10  other  equal  parts, 
which  may  be  called  divisions  of  the  second  order ;  and  each 
of  these  is  divided  into  two  equal  parts,  forming  divisions  of 
the  third  order. 

The  divisions  on  all  the  scales  are  contained  between  four 
parallel  lines  :  those  of  the  first  order  extend  to  the  most  dis- 
tant, those  of  the  third  to  the  least,  those  of  the  second  to  the 
intermediate  parallel. 

When  the  whole  line  of  lines  represents  100,  the  divisions 
of  the  first  order,  or  those  to  which  the  figures  are  annexed, 
represent  tens  ;  those  of  the  second  order,  units  ;  those  of  the 
third  order,  the  halves  of  these  units.  If  the  whole  line  repre- 


70  MATHEMATICAL 

sents  ten,  then  the  divisions  of  the  first  order  are  units  ;  those 
of  the  second,  tenths  ;  and  the  third,  twentieths. 

In  the  line  of  tangents,  the  divisions  to  which  the  numbers 
are  affixed  are  the  degrees  expressed  by  those  numbers.  Every 
fifth  degree  is  denoted  by  a  line  somewhat  longer  than  the  rest ; 
between  every  number  and  each  fifth  degree  there  ar  four 
divisions  longer  than  the  intermediate  adjacent  ones  ;  these  are 
whole  degrees ;  the  shorter  ones,  or  those  of  the  third  order,  are 
30  minutes. 

From  the  centre  to  60  degrees  the  line  of  sines  is  divided 
like  the  line  of  tangents ;  from  60  to  70  it  is  divided  only  to 
every  degree ;  from  70  to  80  to  every  two  degrees  ;  from  80 
to  90  the  division  must  be  estimated  by  the  eye. 

The  divisions  on  the  line  of  chords  are  to  be  estimated  in 
the  same  manner  as  the  tangents. 

The  lesser  line  of  tangents  is  graduated  every  two  degrees 
from  45  to  50  ;  but  from  50  to  60  to  every  degree ;  from  60 
to  the  end  to  half-degrees. 

The  line  of  secants  from  0  to  10  is  to  be  estimated  by  the 
eye ;  from  20  to  50  it  is  divided  to  every  two  degrees ;  from 
50  to  60  to  every  degree ;  and  from  60  to  the  end  to  every 
half-degree. 

The  solution  of  questions  on  the  sector  is  said  to  be  simple 
when  the  work  is  begun  and  ended  on  the  same  line  ;  com- 
pound when  the  operation  begins  on  one  line  and  is  finished 
on  the  other. 

The  operation  varies  also  by  the  manner  in  which  the  com- 
passes are  applied  to  the  sector.  If  a  measure  be  taken  on 
any  of  the  sectoral  lines  beginning  at  the  centre,  it  is  called  a 
lateral  distance.  But  if  the  measure  be  taken  from  any  point 
in  one  line  to  its  corresponding  point  on  the  line  of  the  same 
denomination  on  the  other  leg,  it  is  called  a  transverse  or  par- 
allel distance. 

The  divisions  of  each  sectoral  line  are  bounded  by  three 
parallel  lines  ;  the  innermost  of  these  is  that  on  which  the 
points  of  the  compasses  are  to  be  placed,  because  this  alone  is 
the  line  which  goes  to  the  centre,  and  is  alone,  therefore,  the 
sectoral  line. 

We  shall  now  proceed  to  give  a  few  general  instances  of  the 
manner  of  operating  with  the  sector. 

Multiplication  by  the  line  of  lines.  Make  the  lateral  dis- 
tance of  one  of  the  factors  the  parallel  distance  of  10 ;  then  the 
parallel  distance  of  the  other  factor  is  the  product. 

Example.     Multiply  5  by  6 :  extend  the  compasses  from  the 


DRAWING  INSTRUMENTS.  71 

centre  of  the  sector  to  5  on  the  primary  divisions,  and  open  the 
sector  till  this  distance  become  the  parallel  distance  from  10  to 
10  on  the  same  divisions ;  then  the  parallel  distance  from  6 
to  6,  extended  from  the  centre  of  the  sector,  shall  reach  to 
3,  which  is  now  to  be  reckoned  30.  At  the  same  opening 
of  the  sector,  the  parallel  distance  of  7  shall  reach  from  the 
centre  to  35,  that  of  8  shall  reach  from  the  centre  to  40,  &c. 

Division  by  the  line  of  lines.  Make  the  lateral  distance  of 
the  dividend  the  parallel  distance  of  the  divisor,  the  parallel 
distance  of  1 0  is  the  quotient.  Thus,  to  divide  30  by  5,  make 
the  lateral  distance  of  30,  viz.  3  on  the  primary  divisions,  the 
parallel  distance  of  5  of  the  same  divisions  ;  then  the  parallel 
distance  of  10,  extended  from  the  centre,  shall  reach  to  6. 

Proportion  by  the  line  of  lines.  Make  the  lateral  dis- 
tance of  the  second  term  the  parallel  distance  of  the  first  term ; 
the  parallel  distance  of  the  third  term  is  the  fourth  proportional. 

Example.  To  find  a  fourth  proportional  to  8,  4,  and  6,  take 
the  lateral  distance  of  4,  and  make  it  the  parallel  distance  of  8, 
then  the  parallel  distance  of  6,  extended  from  the  centre,  shall 
reach  to  the  fourth  proportional  3. 

In  the  same  manner  a  third  proportional  is  found  to  two  num- 
bers. Thus,  to  find  a  third  proportional  to  8  and  4,  the  sector 
remaining  as  in  the  former  example,  the  parallel  distance  of  4, 
extended  from  the  centre,  shall  reach  to  the  third  proportional 

2.  In  all  these  cases,  if  the  number  to  be  made  a  parallel  dis- 
tance be  too  great  for  the  sector,  some  aliquot  part  of  it  is  to 
be  taken,  and  the  answer  multiplied  by  the  number  by  which 
the  first  number  was  divided.     Thus,  if  it  were  requioj^l  to  find 
a  fourth  proportional  to  4, 8,  and  6,  because  the  lateraHh'stance 
of  the  second  term  8  cannot  be  made  the  parallel  distance  of 
the  first  term  4,  take  the  lateral  distance  of  4,  viz.  the  half  of 

'  8,  and  make  it  the  parallel  distance  of  the  first  term  4  ;  then 
the  parallel  distance  of  the  third  term  6  shall  reach  from  the 
centre  to  6>  viz.  the  half  of  12.  Any  other  aliquot  part  of  a 
number  may  be  used  in  the  same  way.  In  like  manner,  if  the 
number  proposed  be  too  small  to  be  made  the  parallel  distance, 
it  may  be  multiplied  by  some  number,  and  the  answer  is  to  be 
divided  by  the  same  number. 

To  protract  angles  by  the  line  of  chords.  Case  1 .  When 
the  given  degrees  are  under  60.  1.  With  any  radius  on  a 
centre,  describe  the  arch.  2.  Make  the  same  radius  a  trans- 
verse distance  between  60  and  60  on  the  same  line  of  chords. 

3.  Take  out  the  transverse  distance  of  the  given  degrees,  and 
lay  this  on  the  arch,  which  will  mark  out  the  angular  distance 
required. 


72  MATHEMATICAL 

Case  2.  When  the  given  degrees  are  more  than  60.  1.  Open 
th  sector,  and  describe  the  arch  as  before.  2.  Take  £  or  ±  of 
the  given  degrees,  and  take  the  transverse  distance  of  this  ^  or 
£,  and  lay  it  off  twice  if  the  degrees  were  halved,  three  times  if 
the  third  was  used  as  a  transverse  distance. 

Case  3.  When  the  required  angle  is  less  than  6  degrees ; 
suppose  3.  1.  Open  the  sector  to  the  given  radius,  and  de- 
scribe the  arch  as  before.  2.  Set  off  the  radius.  3.  Set  off 
the  chord  of  57  degrees  backwards,  which  will  give  the  arc  of 
three  degrees. 

Given  the  radius  of  a  circle  (suppose  equal  to  two  inches),  re' 
quired  the  sine  and  tangent  of  28°  30'  to  that  radius. 

Solution. — Open  the  sector  so  that  the  transverse  distance 
of  90  and  90  on  the  sines,  or  of  45  and  45  on  the  tangents, 
may  be  equal  to  the  given  radius,  viz.  two  inches ;  then  will 
the  transverse  distance  of  28°  30',  taken  from  the  sines,  be  the 
length  of  that  sine  to  the  given  radius ;  or  if  taken  from  the 
tangents,  will  be  the  length  of  that  tangent  to  the  given  radius. 
,  But  if  the  secant  of  28°  30'  was  required  ? 

Make  the  given  radius,  two  inches,  a  transverse  distance  to 
0  and  0  at  the  beginning  of  the  line  of  secants ;  and  then  take 
the  transverse  distance  of  the  degrees  wanted,  viz.  28°  30'. 

A  tangent  greater  than  45s  (suppose  60°)  is  found  thus. 

Make  the  given  radius,  suppose  two  inches,  a  transverse  dis- 
tance to  45  and  45  at  the  beginning  of  the  scale  of  upper  tan- 
gents ;  and  then  the  required  number  60°  may  be  taken  from 
this  scale. 

Given  the  length  of  the  sine,  tangent,  or  secant  of  any  degrees, 
to  find  the  length  of  the  radius  to  that  sine,  tangent,  or  secant. 

Solution. — Make  the  given  length  a  transverse  distance  to  its 
given  degrees  on  its  respective  scale  :  then, 

In  the  sines.  The  transverse  distance  of  90  and  90  will  be 
the  radius  sought., 

In  the  lower  tangents.  The  transverse  distance  of  45  and 
45,  near  the  end  of  the  sector,  will  be  the  radius  sought. 

In  the  upper  tangents.  The  transverse  distance  of  45  and 
45,  taken  towards  the  centre  of  the  sector  on  the  line  of  upper 
tangents,  will  be  the  centre  sought. 

In  the  secant.  The  transverse  distance  of  0  and  0,  or  the 
beginning  of  the  secants,  near  the  centre  of  the  sector,  will  be 
the  radius  sought. 

Given  the  radius  and  any  line  representing  a  sine,  tangent,  or 
secant,  to  find  the  degrees  corresponding  to  that  line. 

Solution. — Set  the  sector  to  the  given  radius,  according  as  a 
sine,  or  tangent,  or  secant  is  concerned. 


DRAWING  INSTRUMENTS.  73 

Take  the  given  line  between  the  compasses  ;  apply  the  two 
feet  transversely  to  the  scale  concerned,  and  slide  the  feet  along 
till  they  both  rest  on  like  divisions  on  both  legs ;  then  will  those 
divisions  show  the  degrees  and  parts  corresponding  to  the 
given  line. 

To  find  the  length  of  a  versed  sine  to  a  given  number  of  de- 
grees, and  a  given  radius. 

Make  the  transverse  distance  of  90  and  90  on  kthe  sines 
equal  to  the  given  radius. 

Take  the  transverse  distance  of  the  sine  complement  of  the 
given  degrees. 

If  the  given  degrees  are  less  than  90,  the  difference  between 
the  sine  complement  and  the  radius  gives  the  versed  sine. 

If  the  given  degrees  are  more  than  90,  the  sum  of  the  sine 
complement  and  the  radius  gives  the  versed  sine. 

To  open  the  legs  of  the  sector  so  that  the  corresponding  double 
scales  of  lines  chords,  sines,  and  tangents  may  make  each  a  right 
angle. 

On  the  lines,  make  the  lateral  distance  10  a  distance  be- 
tween 8  on  one  leg  and  6  on  the  other  leg. 

On  the  sines,  make  the  lateral  distance  90  a  transverse  dis- 
tance from  45  to  45 ;  or  from  40  to  50  ;  or  from  30  to  60 ;  or 
from  the  sine  of  any  degrees  to  their  complement  I 

Or  on  the  sines,  make  the  lateral  distance  of  45  a  transverse 
distance  between  30  and  30. 

OF    THE    PLAIN    SCALE. 

The  divisions  laid  down  on  the  plain  scale  are  of  two  kinds, 
the  one  having  more  immediate  relation  to  the  circle  and  its 
properties,  the  other  being  merely  concerned  with  dividing 
straight  lines. 

Though  arches  of  a  circle  are  the  most  natural  measure  of 
an  angle,  yet  in  many  cases  right  lines  are  substituted,  as  being 
more  convenient ;  for  the  comparison  of  one  right  line  with 
another  is  more  natural  and  easy  than  the  comparison  of  a 
right  line  with  a  curve  :  hence  it  is  usual  to  measure  the  quan- 
tities of  angles,  not  by  the  arch  itself,  which  is  described  on  the 
angular  point,  but  by  certain  lines  described  about  that  arch. 

The  lines  laid  down  on  the  plain  scales  for  the  measuring  of 
angles,  or  the  protracting  scales,  are,  1.  A  line  of  chords  marked 
CHO.  2.  A  line  of  sines  marked  SIN.,  of  tangents  marked  TAN., 
of  semitangents  marked  ST.,  and  of  secants  marked  SEC.  ;  this 
last  'is  often  upon  the  same  line  as  the  sines,  because  its  gra- 
dations do  not  begin  till  the  sines  end. 

D 


74  MATHEMATICAL 

There  are  two  other  scales,  namely,  the  rhumbs  marked  RV, 
'and  longitudes  marked  LON.  Scales  of  latitude  and  hours  are 
sometimes  put  upon  the  plain  scale ;  but  as  dialling  is  now  but 
seldom  studied,  they  are  only  made  to  order. 
j  The  divisions  used  for  measuring  straight  lines  are  called 
scales  of  equal  parts,  and  are  of  various  lengths  for  the  conve- 
nience of  delineating  any  figure  of  a  larger  or  smaller  size,  ac- 
cording to  the  fancy  or  purposes  of  the  draughtsman.  They  are, 
indeed,  nothing  more  than  a  measure  in  miniature  for  laying  down 
upon  paper,  &c.  any  known  measure,  as  chains,  yards,  feet,  &c., 
each  part  on  the  scale  answering  to1  one  foot,  one  yard,  &c.,  and 
the  plan  will  be  larger  or  smaller  as  the  scale  contains  a  smaller 
or  a  greater  number  of  parts  in  an  inch.  Hence  a  variety  of 
scales  is  useful  to  lay  down  lines  of  any  required  length,  and 
of  a  convenient  proportion  with  respect  to  the  size  of  the  dr^aw- 
ing.  If  none  of  the  scales  happen  to  suit  the  purpose,  re- 
course should  be  had  to  the  line  of  lines  on  the  sector ;  for,  by 
the  different  openings  of  that  .instrument,  a  line  of  any  length 
may  be  divided  into  as  many 'equal  parts  as  any  person  chooses. 

Scales  of  equal  parts  are  divided  into  two  kinds,  the  one 
simply,  the  other  diagonally  divided. 

Six  of  the  simply  divided  scales  are  generally  placed  one 
above  another  upon  the  same  rule ;  they  are  divided  into  as 
many  equal  parts  as  the  length  of  the  rule  will  admit  of;  -the 
numbers  placed  on  the  right-hand  show  how  many  parts  in  an 
inch  each  scale  is  divided  into.  The  upper  scale  is  sometimes 
shortened  for  the  sake  of  introducing  another,  called  the  line  of 
chords. 

The  first  of  the  larger  or  primary  divisions  on  every  scale 
is  subdivided  into  ten  equal  parts,  which  small  parts  are 
those  which  give  a  name  to  the  scale  :  thus  it  is  called  a  scale 
of  20,  when  20  of  these  divisions  are  equal  to  one  inch.  If, 
therefore,  these  less  divisions  be  taken  as  units,  and  each  repre- 
sents one  league,  one  mile,  one  chain,  or  one  yard,  &c.,  then 
will  the  larger  divisions  be  so  many  tens;  but  if  the  sub- 
divisions are  supposed  to  be  tens,  the  larger  divisions  will  be 
hundreds. 

!  To  illustrate  this,  suppose  it  were  required  fo  set  off  from 
either  of  the  scales  of  equal  parts  f f,  36,  or  360  parts,  either 
miles  or  leagues.  Set  one  foot  of  your  compasses  on  3,  among 
the  larger  or  primary  divisions,  and  open  the  other  point  till  it 
falls  on  the  sixth  subdivision,  reckoning  backwards  or  towards 
the  left  hand.  Then  will  this  extent  represent  f  £ ,  36,  or  360 
miles  or  leagues,  &c.  and  bear  the  same  proportion  in  the  plan 
as  the  line  measured  does  to  the  thing  represented. 


DRAWING  INSTRUMENTS.  75 

To  adapt  these  scales  to  feet  and  inches,  the  first  primary 
division  is  often  duodecimally  divided  by  the  upper  line  ;  there- 
fore, to  lay  down  any  number  of  feet  and  inches,  as,  for  in- 
stance, 8  feet  8  inches,,  exte'nd  the  compasses  from  8  of  the 
larger  to  8  of  the  upper  small  ones,  and  that  distance  laid  down 
on  the  plan  will  represent  8  feet  8  inches. 

Of  the  scale  of  equal  parts  diagonally  divided.  The  use  of 
this  scale  is  the  same  as  those  already  described.  But  by  it  a 
plane  may  be  more  accurately  divided  than  by  the  former ;  for 
any  one  of  the  larger  divisions  may  by  this  be  subdivided  into 
100  equal  parts  ;  and,  therefore,  if  the  scale  contains  10  of  the 
larger  divisions,  any  number  under  1000  may  be  laid  down  with 
accuracy. 

The  diagonal  scale  is  seldom  placed  on  the  same  side  of  the 
rule  with  the  other  plotting  scale.  The  first  division  of  the 
diagonal  scale,  if  it  be  a  foot  long,  is  generally  an  inch  divided 
into  100  equal  parts,  and  at  the  opposite  there  is  usually  half 
an  inch  divided  into  100  equal  parts.  If  the  scale  be  six  inches 
long,  one  end  has  commonly  half  an  inch,  the  other  a  quarter 
of  an  inch,  subdivided  into  100  equal  parts. 

The  nature  of  this  scale  will  be  better  understood  by  consider- 
ing its  construction.  For  this  purpose, 

First.  Draw  eleven  parallel  lines  at  equal  distances  ;  divide 
the  upper  of  these  lines  into  such  a  number  of  equal  parts  as 
the  scale  to  be  expressed  is  intended  to  contain ;  from  each  of 
these  divisions  draw  perpendicular  lines  through  the  eleven 
parallels. 

Secondly.  Subdivide  the  first  of  these  divisions  into  ten 
equal  parts,  both  in  the  upper  and  lower  lines. 

Thirdly.  Subdivide  again  each  of  these  subdivisions,  by 
drawing  diagonal  lines  from  the  1  Oth  below  to  the  9th  above  ; 
from  the  8th  below  to  the  7th  above  ;  and  so  on,  till  from  the 
first  below  to  the  0  above ;  by  these  lines  each  of  the  small 
divisions  is  divided  into  ten  parts,  and  consequently  the  whole 
first  space  into  1 00  equal  parts  ;  for  as  each  of  the  subdivisions 
is  one-tenth  part  of  the  whole  first  space  or  division,  so  each 
parallel  above  it  is  one-tenth  of  such  subdivision,  and  conse- 
quently, one-hundredth  part  of  the  whole  first  space ;  and  if 
there  be  ten  of  the  larger  divisions,  one  thousandth  part  of  the 
whole  space. 

If,  therefore,  the  larger  divisions  be  accounted  as  units,  the 
first  subdivisions  will  be  tenth  parts  of  a  unit,  and  the  second, 
marked  by  the  diagonal  upon  the  parallels,  hundredth  parts  of 
the  unit  But  if  we  suppose  the  larger  divisions  to  be  tens, 
the  first-  subdivisions  will  be  units  and  the  second  tenths.  If 
D2 


76  MATHEMATICAL 

the  larger  are  hundreds,  then  will  the  first  be  tens  and  the 
second  units. 

The  numbers,  therefore,  576,  57,6,  5,76,  are  all  expressible 
by  the  same  extent  of  the  compasses  :  thus,  setting  one  foot  in 
the  number  5  of  the  larger  divisions,  extend  the  other  along 
the  sixth  parallel  to  the  seventh  diagonal.  For,  if  the  five 
larger  divisions  be  -taken  for  500,  seven  of  the  first  subdivisions 
will  be  70,  which  upon  the  sixth  parallel,  taking  in  six  of  the 
second  subdivisions  for  units,  makes  the  whole  number  576. 
Or,  if  the  five  larger  divisions  be  taken  for  five  tens,  or  50, 
seven  of  the  first  subdivisions  will  be  seven  units,  and  the  six 
second  subdivisions  upon  the  sixth  parallel  will  be  six  tenths 
of  a  unit.  Lastly,  if  the  five  larger  divisions  be  only  esteemed 
as  five  units,  then  will  the  seven  first  subdivisions  be  seven 
tenths,  and  the  six  second  subdivisions  be  the  six  hundredth 
parts  of  a  unit. 

Of  the  line  of  chords.  This  line  is  used  to  set  off  an  angle 
from  a  given  point  in  any  right  line,  or  to  measure  the  quan- 
tity of  an  angle  already  laid  down. 

Thus,  to  draw  a  line  that  shall  make  with  another  line  an 
angle  containing  a  given  number  of  degrees,  suppose  40  de- 
grees. 

Open  your  compasses  to  the  extent  of  60  degrees  upon  the 
line  of  chords  (which  is  always  equal  to  the  radius  of  the  circle 
of  projection),  and  setting  one  foot  in  the  angular  point,  with 
that  extent  describe  an  arch  ;  then  taking  the  extent  of  40  de- 
grees from  the  said  chord  line,  set  it  off  from  the  given  line  on 
the  arch  described ;  a  right  line  drawn  from  the  given  point 
through  the  point  marked  upon  the  arch  will  form  the  required 
angle. 

\  The  degrees  contained  in  an  angle  already  laid  down  are 
found  nearlyin  the  same  manner.  For  instance,  to  measure  an 
angle :  from  the  centre  describe  an  arch  with  the  chord  of 
60  degrees,  and  the  length  of  the  arch  contained  between  the 
lines  measured  on  the  line  of  chords  will  give  the  number  of 
degrees  contained  in  the  angle. 

If  the  number  of  degrees  are  more  than  90,  they  must  be 
measured  upon  the  chords  at  twice :  thus,  if  120  degrees  were 
to  be  practised,  60  may  be  taken  from  the  chords,  and  those 
degrees  be  laid  off  twice  upon  the  arch.  Degrees  taken  from  the 
chords  are  always  to  be  counted  from  the  beginning  of  the  scale. 

Of  the  rhumb  line.  This4  is,  in  fact,  a  line  of  chords  con- 
structed to  a  quadrant  divided  into  eight  parts  or  points  of  the 
compass,  in  order  to  facilitate  the  work  of  the  navigator  in  lay- 
ing down  a  ship's  course. 


DRAWING  INSTRUMENTS.  77 

Of  the  line  of  longitudes.  The  line  of  longitudes  is  a  line 
divided  into  sixty  unequal  parts,  and  so  applied  to  the  line  of 
chords  as  to  show,  by  inspection,  the  number  of  equatorial 
miles  contained  in  a  degree  on  any  parallel  of  latitude.  ,  The 
graduated  line  of  chords  is  necessary,  in  order  to  show  the 
latitudes ;  the  line  of  longitude  shows  the  quantity  of  a  degree 
on  each  parallel  in  sixtieth  parts  of  an  equatorial  degree,  that 
is,  miles. 

The  lines  of  tangents,  semitangents  and  secants  serve  to  find 
the  centres  and  poles  of  projected  circles  in  the  stereographical 
projection  of  the  sphere. 

The  line  of  sines  is  principally  used  for  the  orthographic 
projection  of  the  sphere. 

The  lines  of  latitudes  and  hours  are  used  conjointly,  and 
serve  very  readily  to  mark  the  hour  lines  in  the  construction 
of  dials  :  they  are  generally  on  the  most  complete  sorts  of  scales 
and  sectors;  for  the  uses  of  which  see  treatises  on  dialling. 


OF    THE    PROTRACTOR. 

This  is  an  instrument  used  to  protract  or  lav  down  an  angle 
containing  any  number  of  degrees,  or  to  find  how  many  degrees . 
are  contained  in  any  given  angle.  There  are  two  kinds  put 
into  cases  of  mathematical  drawing  instruments ;  one  in  the 
form  of  a  semicircle,  the  other  in  the  form  of  a  parallelogram. 
The  circle  is  undoubtedly  the  only  natural  measure  of  angles ; 
when  a  straight  line  is  therefore  used  the  divisions  thereon  are 
derived  from  a  circle  or  its  properties,  and  the  straight  line  is 
made  use  of  for  some  relative  convenience  :  it  is  thus  the  par- 
allelogram is  often  used  as  a  protractor  instead  of  the  semi- 
circle, because  it  is  in  some  cases  more  convenient,  and  that 
other  scales,  <fcc.  may  be  placed  upon  it. 

The  semicircular  protractor  is  divided  into  1 80  equal  parts 
or  degrees,  which  are  numbered  at  every  tenth  degree  each 
way,  for  the  conveniency  of  reckoning  either  from  the  right 
towards  the  left,  or  from  the  left  towards  the  right ;  or  the  more 
easily  to  lay  down  an  angle  from  either  end  of  the  line,  begin- 
ning at  each  end  with  10,  20,  &c.  and  proceeding  to  180  de- 
grees. The  edge  is  the  diameter  of  the  semicircle,  and  the 
mark  in  the  middle  points  out  the  centre,  in  a  protractor  in  the 
form  of  a  parallelogram :  the  divisions  are,  as  in  the  semicircu- 
lar one,  numbered  both  ways ;  the  blank  side  represents  the 
diameter  of  a  circle.  The  side  of  the  protractor  to  be  applied 
to  the  paper  is  made  flat,  and  that  whereon  the  degrees  are 
marked  is  chamfered  or  sloped  away  to  the  edge,  that  an  angle 


78  MATHEMATICAL 

may  be  more  easily  measured,  and  the  divisions  set  off  with 
greater  exactness. 

Application  of  the  protractor  to  use.  \.  A  number  of  degrees 
being  *given,  to  protract  or  lay  down  an  angle  whose  measure 
shall  be  equal  thereto. 

Thus,  to  lay  down  an  angle  of  60  degrees  from  the  point  of 
a  line,  apply  the  diameter  of  the  protractor  to  the  line,  so  that 
the  centre  thereof  may  coincide  exactly  with  the  extremity ; 
then,  with  a  protracting*  pin  make  a  fine  dot  against  60  upon 
the  limb  of  the  protractor ;  now  remove  the  protractor,  and  draw 
a  line  from  the  extremit)*-  through  that  point,  and  the  angle  con- 
tains the  given  number  of  degrees. 

2.  To  find  the  number  of  degrees  contained  in  a  given  angle* 
Place  the  centre  of  the  protractor  upon  the  angular  point,  and 

the  fiducial  edge  or  diameter  exactly  upon  the  line ;  then  the 
degree  upon  the  limb  that  is  cut  by  the  line  will  be  the  measure 
of  the  given  angle,  which,  in  the  present  instance  is  found  ta 
be  60  degrees. 

3.  From  a  given  point  in  a  line  to  erect  a  perpendicular  to  that 
line. 

Apply  the  protractor  to  the  line,  so  that  the  centre  may  co- 
incide with  the  given  point,  and  the  division  marked  90  may  be 
out  by  the  line  ;  then  a  line  drawn  against  the  diameter  of  the 
protractor  will  be  the  perpendicular  required. 

OF  PARALLEL  RULES. 

Parallel  lines  occur  so  continually  in  every  species  of  mathe- 
matical drawing,  that  it  is  no  wonder  so  many  instruments 
have  been  contrived  to  delineate  them  with  more  expedition 
than  could  be  effected  by  the  general  geometrical  methods. 
For  this  purpose  rules  of  various  constructions  have  been  made, 
and  particularly  recommended  by  their  inventors ;  their  use, 
however,  is  so  apparent  as  ta  need  no  explanation. 

GUNTER'S  SCALE.. 

The  scale  generally  used  is  a  ruler  two  feet  in  length,  hav- 
ing drawn  upon  it  equal  parts,  chords,  sines,  tangents,  secants, 
&c.  These  are  contained  on  one  side  of  tfie  scale,  and  the 
other  side  contains  the  logarithms  of  these  numbers.  Jfcfrv 
Edmund  Gunter  was  the  first  who  applied  the  logarithms 
of  numbers  and  of.  sines  and  tangents  to  straight  lines  drawn 
on  a  scale  or  ruler,  with  which  proportions  in  common  numbers 
and  trigonometry  may  be  solved  by  the  application  of  a  pair  of 
compasses  only.  The  method  is  founded  on  this  property, 


DRAWING  INSTRUMENTS.  79 

That  the  logarithms  of  the  terms  of  equal  ratios  are  equidifferent. 
This  was  called  Gunter's  Proportion  and  Gunter's  Line ; 
hence  the  scale  is  generally  called  the  Gunter. 

Of  the  Logarithmical  Lines  on  Gunter1  s  Scale. 

The  logarithmical  lines  on  Gunter's  Scale  are  the  eight  fol- 
lowing :  i 

S^Rhumb,  or  fine  rhumbs,  is  a  line  containing  the  logarithms 
of  the  natural  sines  of  every  point  and  quarter  point  of  the 
compass,  numbered  from  a  brass  pin  on  the  right-hand  towards 
the  left  with  8, 7,  6,  5,  4,  3,  2,  1. 

T^Rhumb,  or  tangent  rhumbs,  also  corresponds  to  the  log- 
arithm of  the  tangent  of  every  point  and  quarter  point  of  the 
compass.  This  line  is  numbered  from  near  the  middle  of  the 
scale  with  1,  2,  3,  4,  towards  the  right-hand,  and  back  again 
with  the  numbers  5,  6,  7,  from  the  right-hand  towards  the  left. 
To  take  off  any  number  of  points  below  4,  we  irlust  begin  at 
1  and  count  towards  the  right-hand  ;  but  to  take  off  any  num- 
ber of  points  above  4,  we  must  begin  at  4  and  count  towards 
the  left-hand. 

Numbers,  on  the  line  of  numbers,  is  numbered  from  the  left- 
hand  of  the  scale  towards  the  right,  with  1,  2,  3,  4,  5,  6,  7,  8, 
9,  1,  which  stands  exactly  in  the  middle  of  the  scale ;  the  num- 
bers then  go  on  2,  3,  4,  5,  6,  7,  8,  9,  10,  which  stands  at  the 
right-hand  end  of  the  scale.  These  two  equal  parts  of  the 
scale  are  divided  equally,  the  distance  between  the  first  or  left- 
hand  1  and  the  first  2,  3,  4,  &c.  is  exactly  equal  to  the  distance 
between  the  middle  1  and  numbers  2,  3,  4,  &c.  which  follow 
it.  The  subdivisions  of  these  scales  are  likewise  similar,  viz. 
they  are  each  one-tenth  of  the  primary  divisions,  and  are  dis- 
tinguished by  lines  of  about  half  the  length  of  the  primary 
divisions. 

These  subdivisions  are  again  divided  into  ten  parts,  where 
room  will  permit ;  and  where  that  is  not  the  case  the  units  must 
be  estimated  or  guessed  at  by  the  eye,  which  is  easily  done  by 
a  little  practice. 

The  primary  divisions  on  the  second  part  of  the  scale  are 
estimated  according  to  the  value  set  upon  the  unit  on  the  left- 
hand  of  the  scale :  if  you  call  it  one,  then  the  first  1,  2,  3,  &c. 
stand  for  1,  2,  3,  &c. ;  the  middle  1  is  10,  and  the  2,  3,  4,  &c. 
following  stand  for  20,  30,  40,  &c. ;  and  the  10  at  the  right- 
hand  is  100.  If  the  first  1  stand  for  10,  the  first  2,  3,  4,  <fcc. 
must  be  counted  20,  30,  40,  &c. ;  the  middle  1  will  be  100, 
and  the  second  2, 3, 4,  5,  &c.  will  stand  for  200,  300,  400,  500, 
&c. ;  and  the  10  at  the  right  for  1000. 

If  you  consider  the  first  1  as  ^  of  a  unit,  the  2,  3,  4,  &c. 


80  MATHEMATICAL 

following  will  be  •&,  f^,  T4¥,  &c. ;  the  middle  1  will  stand  for 
a  unit,  and  the  2, 3, 4,  &c.  following  will  stand  for  2,  3, 4,  &c.  ; 
also,  the  division  at  the  right-hand  end  of  the  scale  will  stand 
for  10.  The  intermediate  small  divisions  must  be  estimated 
according  to  the  value  set  upon  the  primary  ones. 

Sine.  The  line  of  sines  is  numbered  from  the  left-hand  of 
the  scale  towards  the  right,  1,  2,  3,  4,  5,  &c.  to  10 ;  then  20r 
30,  40,  &c.  to  90,  where  it  terminates  just  opposite  10  on  the 
line  of  numbers. 

Versed  sine.  This  line  is  placed  immediately  under  the  line 
of  sines,  and  numbered  in  a  contrary  direction,  viz.  from  the 
right-hand  towards  the  left,  10,  20,  30,  40,  50,  to  about  160 ; 
the  small  divisions  are  here  to  be  estimated  according  to  the 
number  of  them  to  a  degree.  By  comparing  the  line  of  versed 
sines  with  the  line  of  sines,  it  will  appear  that  the  versed  sines 
do  not  belong  to  the  arches  with  which  they  are  marked,  but 
are  the  half  versed  sines  of  their  supplements.  Thus,  what  is 
marked  the  versed  sine  of  90  is  only  half  the  versed  sine  of 
90,  the  versed  sine  of  120°  is  half  the  versed  sine  of  60°, 
and  the  versed  sine  marked  100°  is  half  the  versed  sine  of 
80°,  &c. 

The  versed  sines  are  numbered  in  this  manner  to  render 
them  more  commodious  in  the  solution  of  trigonometrical  and 
astronomical  problems. 

Tangents.  The  line  of  tangents  begins  at  the  left-hand,  and 
is  numbered  1,  2,  3,  &c.  to  10,  then  20,  30,  45,  where  there  is 
a  little  brass  pin  just  under  90  in  the  line  of  sines,  because  the 
sine  of  90°  is  equal  to  the  tangent  of  45°.  It  is  numbered  from 
45°  towards  the  left-hand  50,  60,  70,  80,  &c.  The  tangents 
of  arches  above  45°  are  therefore  counted  backward  on  the 
line,  and  are  found  at  the  same  points  of  the  line  as  the  tan- 
gents of  their  complements. 

Thus  the  division  at  40  represents  both  40  and  50,  the  di- 
vision at  30  serves  for  30  and  60,  &c. 

Meridional  Parts.  This  line  stands  immediately  above  a 
line  of  equal  parts,  marked  Equal  Pt.,  with  which  it  must  always 
be  compared  when  used.  The  line  of  equal  parts  is  marked 
from  the  right-hand  to  the  left  with  0, 10, 20, 30,  &c. ;  each  of 
these  large  divisions  represents  10  degrees  of  the  equator, 
or  600  miles.  The  first  of  these  divisions  is  sometimes  di- 
vided into  40  equal  parts,  each  representing  15  minutes  or 
miles. 

The  extent  from  the  brass  pin  on  the  scale  of  meridional 
parts  to  any  division  on  that  scale,  applied  to  the  line  of  equal 
parts,  will  give  (in  'degrees)  the  meridional  parts  answering  to 


DRAWING  INSTRUMENTS,  81 

the  latitude  of  that  division.  Or  the  extent  from  any  division 
to  another  on  the  line  of  meridional  parts,  applied  to  the  line 
of  equal  parts,  will  give  the  meridjonal  difference  of  latitude  be- 
tween the  two  places  denoted  by  the  divisions.  These  degrees 
are  reduced  to  leagues  by  multiplying  by  20,  or  to  miles  by 
multiplying  by  60. 

The  use  of  the  Logarithmical  Lines  on  Gunter's  Scale. 
By  these  lines  and  a  pair  of  compasses  all  the  problems  of 
trigonometry,  <fcc.  may  be  solved. 

These  problems  are  all  solved  by  proportion.  Now,  in  natu- 
ral numbers  the  quotient  of  the  first  term  by  the  second  is 
equal  to  the  quotient  of  the  third  by  the  fourth :  therefore,  loga- 
rithmically speaking,  the  difference  between  the  first  and  sec- 
ond term  is  equal  to  the  difference  between  the  third  and  fourth ; 
consequently,  on  the  lines  on  the  scale  the  distance  between  the 
first  and  second  term  will  be  equal  to  the  distance  between  the 
third  and  fourth.  And  for  a  similar  reason,  because  four  pro- 
portional quantities  are  alternately  proportional,  the  distance 
between  the  first  and  third  terms  will  be  equal  to  the  distance 
between  the  second  and  fourth.  Hence  the  following 

General  Rule. 

The  extent  of  the  compasses  from  the  first  term  to  the  second 
will  reach,  in  this  same  direction,  from  the  third  to  the  fourth 
term.  Or,  the  extent  of  the  compasses  from  the  first  term  to 
the  third  will  reach,  in  the  same  direction,  from  the  second  to 
the  fourth. 

By  the  same  direction  in  the  foregoing  rule  is  meant,  that  if 
the  second  term  lie  on  the  right-hand  of  the  first  the  fourth  will 
lie  on  the  right-hand  of  the  third,  and  the  contrary.  This  is 
true,  except  the  two  first  or  two  last  terms  of  the  proportion  are 
on  the  line  of  tangents,  and  neither  of  them  under  45°  ;  in  this 
case,  the  extent  on  the  tangents  is  to  be  made  in  a  contrary 
direction :  for  had  the  tangents  above  45°  been  laid  down  in 
their  proper  direction,  they  would  have  extended  beyond  the 
length  of  the  scale  towards  the  right-hand  ;  they  are  therefore, 
as  it  were,  folded  back  upon  the  tangents  below  45°,  and  con- 
sequently lie  in  a  direction  contrary  to  their  proper  and  natural 
order. 

If  the  two  last  terms  of  a  proportion  be  on  the  line  of  tan- 
gents, and  one  of  them  greater  and  the  other  less  than  45°,  the 
extent  from  the  first  term  to  the  second  will  reach  from  the 
third  beyond  the  scale.  To  remedy  this  inconvenience,  apply 
the  extent  between  the  two  first  terms  from  45°  backward 
upon  the  line  of  tangents,  and  keep  the  left-hand  point  of  the 
D  3 


82  TRIGONOMETRY. 

compasses  where  it  falls ;  bring  the  right-hand  pom*  from  45° 
to  the  third  term  of  the  proportion  ;  this  extent  now  in  the  com- 
passes applied  from  45°  backward  will  reach  to  the  fourth  term, 
or  the  tangent  required.  For,  had  the  line  of  tangents  been 
continued  forward  beyond  45°,  the  divisions  would  have  fallea 
above  45°  forward,  in  the  same  manner  as  they  fall  under  45° 
backward* 


SECTION  V. 

TRIGONOMETRY. 

The  word  Trigonometry,  signifies  the  measuring  of  triangles. 
But  under  this  name  is  generally  comprehended  the  art  of  de- 
termining the  positions  and  dimensions  of  the  several  unknown 
parts  of  extension,  by  means  of  some  parts  which  are  already 
known.  If  we  conceive  the  different  points  which  may  be 
represented  in  any  space  to  be  joined  together  by  right  lines, 
there  are  three  things  offered  for  our  consideration ;  1,  the 
length  of  these  lines ;  2,  the  angles  which  they  form  with  one 
another ;  3,  the  angles  formed  by  the  planes  in  which  these 
lines  are  drawn,  or  are  supposed  to  be  traced.  On  the  com- 
parison of  these  three  objects  depends  the  solution  of  all  ques- 
tions that  can  be  proposed  concerning  the  measure  of  extension 
and  its  parts  ;  and  the  art  of  determining  all  these  things  from 
the  knowledge  of  some  of  them  is  reduced  to  the  solution  of 
these  two  general  questions* 

1.  Knowing  three  of  the  six  parts,  the   sides  and  angles, 
which  constitute  a  rectilineal  triangle,  to  find  the  other  three. 

2.  Knowing  three  of  the  six  parts  which  compose  a  spherical 
triangle,  that  is,  a  triangle  formed  on  the  surface  of  a  sphere  by 
three  arches  of  circles  which  have  their  centre  in  the  centre  of 
the  same  sphere,  to  find  the  other  three. 

The  first  question  is  the  object  of  what  is  called  Plane  Trigo- 
nometry, because  the  six  parts  considered  here  are  in  the 
Same  plane :  it  is  also  denominated  Rectilineal  Trigonometry. 
The  second  question  belongs  to  Spherical  Trigonometry, 
wherein  the  six  parts  are  considered  in  different  planes.  But 
the  only  object  here  is  to  explain  the  solutions  of  the  former 
question,  viz. 

PLANE  TRIGONOMETRY. 

Plane  Trigonometry  is  that  branch  of  Geometry  which 


•v 

TRIGONOMETRY.  83 

teaches  how  to  determine  or  calculate  three  of  the  six  parts  of 
a  rectilineal  triangle,  by  having  the  other  three  parts  given  or 
known.  It  is  usually  divided  into  Right-angled  and  Oblique- 
angled  Trigonometry,  according  as  it  is  applied  to  the  mensura- 
tion of  right  or  oblique-angled  triangles. 

In  every  triangle  or  case  in  trigonometry  three  of  the  parts 
must  be  given,  and  one  of  these  parts  at  least  must  be  a  side ; 
because,  if  the  three  angles  only  were  given,  it  is  obvious  that 
all  similar  triangles  would  answer  the  question. 

RIGHT-ANGLED  PLANE  TRIGONOMETRY. 

PL.  5.  fig.  1. 

1.  In  every  right-angled  plane  triangle  ABC,  if  the  hypothe- 
nuse  AC  be  made  the  radius,  and  with  it  a  circle  or  an  arc  of 
one  be  described  from  each  end,  it  is  plain  (from  def.  20),  that 
BC  is  the  sine  of  the  angle  A,  and  AB  is  the  sine  of  the  angle 
C ;  that  is,  the  legs  are  the  sines  of  their  opposite  angles.* 

*  The  sine  and  co-sine  of  any  number  of  degrees  and  minutes  is  found 
by  the  series  (which  is  given  and  illustrated  in  page  49,  Ryan's  Differ- 
ential and  Integral  Calculus) 

|3 


&c.  for  its  sine,  and  its  co-sine  by  1 • 


•  7- 

In  which  series  the  value  of  a  is  found  thus  :  as  the  number  of  degrees 
or  minutes  in  the  whole  semicircle  is  to  the  degrees  or  minutes  in  the  arc 
proposed,  so  is'3.14159,  &c.  to  the  length  of  the  said  arc,  which  is  the 
value  of  a.  For  example,  let  it  be  required  to  find  the  sine  of  one 
minute  ;  then,  as  10800  (the  minutes  in  180  degrees)  :  1  :  :  3.14159,  &c.  : 
.000290888208665=  the  length  of  an  arc  of  one  minute,  which  is  the 

value  of  a  in  this  case,  and  —  (=— )=.000000000004102,  &c.  Conse- 
quently, .000290888208665— .000000000004102=.000290888204563= 
the  required  sine  of  one  minute. 

Again,  let  it  be  required  to  find  the  sine  and  co-sine  of  five  degrees,  each 
true  to  seven  places  of  decimals.  Here,  .0002908882,  the  length  of  an 
arc  of  one  minute  (found  above),  being  multiplied  by  300,  the  number  of 
minutes  in  5  degrees,  the  product  .08726646  is  the  length  of  an  arc  of  5 
degrees  ;  therefore  in  this  case  we  have 

a  =      .08726646 
a3 
——=—.00011076 

a5 

+120-=  +  .  00000004 
&c.  &c. 

Consequently,  .08715574  =  the  sine  of  5  degrees.    Aboj  —=.00380771, 

a4 

and  -24  =.00000241 ;  hence  1— .00380771  +  .00000241=.996194T=  the 
co-sine  of  5  decrees. 


jfc  84  TRIGONOMETRY 

*** 

'  2;  If  one  leg  JL5  be  made  the  radius,  and  with  it  on  the 
point  A  an  arc  be  described,  then  BC  is  the  tangent  and  AC  is 
the  secant  of  the  angle  A,  by  def,  22  and  25. 

Fig.  3. 

3.  If  BC  be  made  the  radius,  and  an  arc  be  described  with  it 
©n  the  point  C,  then  is  AB  the  tangent  and  AC  is  the  secant 
of  the  angle  C,  as  before. 

Because  the  sine,  tangent,  or  secant  of  any  given  arc  in  one 
circle  is  to  the  sine,  tangent,  or  secant  of  a  like  arc  (or  to  one 
ef  the  like  number  of  degrees)  in  another  circle,  as  the  radius 
of  the  one  is  to  the  radius  of  the  other  ;  therefore  the  sine,  tan- 
gent, or  secant  of  any  arc  is  proportional  to  the  sine,  tangent, 
or  secant  of  a  like  arc,  as  the  radius  of  the  given  arc  is  to 
10.000000,  the  radius  from  whence  the  logarithmic  sines,  tan- 
gents, and  secants  in  most  tables  are  calculated ;  that  is, 

If  A  C  be  made  the  radius,  the  sines  of  the  angles  A  and  C, 
described  by  the  radius  .A  C,  will  be  proportional  to  the  sines 
of  the  like  arcs  or  angles  in  the  circle  that  the  tables  now  men- 
tioned were  calculated  for.  So  if  BC  was  required,  having  the 
angles  and  AB  given,  it  will  be, 

After  the  same  manner  the  sine  and  co-sine  of  any  other  arc  may  be  de- 
rived ;  but  the  greater  the  arc  is  the  slower  the  series  will  converge,  and 
therefore  a  greater  number  of  terms  must  be  taken  to  bring  out  the  conclu- 
sion to  the  same  degree  of  exactness. 

Or,  having  found  the  sine,  the  co-sine  will  be  found  from  it  (by  theo. 

14),  the  co-sine  CL  (plate  1,  fig.  8)  =^/  CH2,  HL2,  or  c  =  </  l—s2. 

For  other  methods  of  constructing  the  canon  of  sines  and  co-sines, 
the  reader  is  referred  to  Hutton's  Mathematics,  Simpson's  Algebra,  &c. 

The  sines  and  co-sines  being  known  or  found  by  the  foregoing  method, 
the  tangents  and  secants  will  be  easily  found  from  the  principle  of  similar 
triangles,  in  the  following  manner : 

In  plate  1,  fig.  8,  where  of  the  arc  BH,  HL  is  the  sine,  CL  or  FH 
the  co-sine,  BK  the  tangent,  CK  the  secant,  DI  the  co-tangent,  and  CI 
the  co-secant,  the  radius  being  CH)  or  CB>  or  CD,  the  three  similar 
triangles  CLH,  CBK,  CDI  give  the  following  proportion  (by  theo.  14) : 

1.  CL  :  LH :  :  CB  :  BK ;  whence  the  tangent  is  known,  being  a  fourth 
proportional  to  the  co-sine,  sine,  and  radius. 

2.  CL:  CH::  CB:CK;  whence  the  secant  is  known,  being  a  third 
proportional  to  the  co-sine  and  radius. 

3.  HL  :  LC  :  :  CD  :  D7;  whence  the  co-tangent  is  known,  being  a 
fourth  proportional  to  the  sine,  co-sine,  and  radius. 

4.  HL  :  HC  :  :  CD  :  C7;  whence   the  co-secant  is  known,  being  a 
third  proportional  to  the  sine  and  radius. 

As  for  the  logarithms,  sines,  tangents,  and  secants  in  the  tables,  they 
are  only  the  logarithms  of  the  natural  sines,  tangents,  and  secants  calcu- 
lated as  above. 


TRIGONOMETRY.  8$ 

Fig.  1. 

As  S.C  :  AB  :  :  S.A  :  EC. 

That  is,  as  the  sine  of  the  angle  C  in  the  tables  is  to  the 
length  of  AB  (or  sine  of  the  angle  C  in  a  circle  whose  radius 
is  A  C),  so  is  the  sine  of  the  angle  A  in  the  tables  to  the  length 
of  BC  (or  sine  of  the  same  angle  in  the  circle  whose  radius 
is  AC). 

In  like  manner  the  tangents  and  secants  represented  by 
making  either  leg  the  radius  will  be  proportional  to  the  tangents 
and  secants  of  a  like  arc,  as  the  radius  of  the  given  arc  is  to 
10.000000,  the  radius  of  the  tables  aforesaid. 

Hence  it  is  plain,  that  if  the  name  of  each  side  of  the  triangle 
be  placed  thereon,  a  proportion  will  arise  to  answer  the  same 
end  as  before  :  thus,  if  AC  be  made  the  radius,  let  the  word 
radius  be  written  thereon ;  and  as  BC  and  AB  are  the  sines 
of  their  opposite  angles,  upon  the  first  let  S.A,  or  sine  of  the 
angle  A,  and  on  the  other  let  S.  C,  or  sine  of  the  angle  C,  be 
written.  Then, 

When  a  side  is  required,  it  may  be  obtained  by  this  propor- 
tion, viz. 

As  the  name  of  the  side  given 
is  to  the  side  given, 

So  is  the  name  of  the  side  required 
to  the  side  required. 

Thus,  if  the  angles  A  and  C  and  the  hypothenuse  AC  were 
given,  to  find  the  sides  ;  the  proportion  will  be 

Fig.  1. 

1.  R  :  AC  :  :  S.A  :  BC. 

That  is,  as  radius  is  to  AC,  so  is  the  sine  of  the  angle  A  to 
BC.  And, 

2.  R  :  AC  :  :  S.C  :  AB. 

That  is,  as  radius  is  to  AC,  so  is  the  sine  of  the  angle  Cto  AB. 
When  an  angle  is  required  we  use  this  proportion,  viz. 
As  the  side  that  is  made  the  radius 

is  to  radius, 
So  is  the  other  given  side 

to  its  name» 

Thus,  if  the  legs  were  given,  to  find  the  angle  A,  and  if  AB 
be  made  the  radius,  it  will  be 

Fig.  2. 
AB:  R::  BC:  T.A. 

That  is,  as  AB  is  to  radius,  so  is  BC  to  the  tangent  of  the 
angle  A. 


88  TRIGONOMETRY. 

After  the  same  manner,  the  sides  or  angles  of  all  rigW£ 
angled  plane  triangles  may  be  found,  from  their  proper  data. 

We  here,  in  plate  4,  give  all  the  proportion  requisite  for 
the  solution  of  the  six  cases  in  right-angled  trigonometry ; 
making  every  side  possible  the  radius* 

In 'the  following  triangles  this  mark  —  in  an  angle  denotes.it 
to  be  known,  or  the  quantity  of  degrees  it  contains  to  be  given; 
and  this  mark '  on  a  side  denotes  its  length  to  be  given  in  feet, 
yards,  perches,  or  miles,  &c.  and  this  mark  °,  either  in  an  angle 
or  on  a  side,  denotes  the  angle  or  side  to  be  required. 

From  these  propositions  it  may  be  observed,  that  to  find  a 
side,  when  the  angles  and  one  side  are  given,  any  side  may  be 
made  the  radius  ;  and  to  find  an  angle,  one  of  the  given  sides 
must  be  made  the  radius.  So  that  in  the  1st,  2d,  and  3d  cases 
any  side,  as  well  required  as  given,  may  be  made  the  radius, 
and  in  the  first  statings  of  the  4th,  5th,  and  6th  cases,  a  given 
side  only  is  made  the  radius. 


RIGHT-ANGLED  TRIANGLES. 

CASE  I. 

The  angles  and  hypothenuse  given,  to  find  the  base  and  perpendicular. 
PL.  5.  fig.  4. 

In  the  right-angled  triangle  ABC,  suppose  the  angle  A= 
46°  30' ;  and  consequently  the  angle  C=43°  30'  (by  cor.  2y 
theo.  5);  and  AC  250  parts  (as  feet,  yards,  miles,  &c.) ;  re- 
quired the  sides  AB  andJ?C. 

1st.     By  Construction* 

Make  an  angle  of  46°  30'  in  blank  lines  (by  prob.  16,  geom.)^ 
as  CAB ;  lay  250,  which  is  the  given  hypothenuse,  from  a 
scale  of  equal  parts,  from  AtoC;  from  C  let  fall  the  perpen- 
dicular BC  (by  prob.  7,  geom.),  and  that  will  constitute  the 
triangle  ABC.  Measure  the  lines  BC  and  AB  from  the  same 
scale  of  equal  parts  that  AC  was  taken  from,  and  you  have 
the  answer.* 

*  It  is  proper  to  observe,  that  constructions,  though  perfectly  correct  in 
theory,  would  give  only  a  moderate  approximation  in  practice j  on  account 
of  the  imperfection  of  the  instruments  required  in  constructing  them ; 
they  are  called  graphic  methods.  Trigonometrical  methods,  on  the  con- 
trary, being  independent  of  all  mechanical  operation,  give  solutions  with 
the  utmost  accuracy :  they  are  founded  upon  the  properties  of  lines  called 
sines,  co-sines,  tangents,  &c.,  which  furnish  a  very  simple  mode  of  express- 
ing the  relations  that  subsist  between  the  sides  and  angles  of  triangles. 


TRIGONOMETRY.  87 

2d.     By  Calculation. 

I.    Making  A  C  the  radius,  the  required  sides  are  found  by 
these  propositions,  as  in  plate  4,  case  t. 

R  :  AC  :  :  S.A  :  BC. 

R:AC::S.C:AB~ 

That  is,  as  radius  =90° 

is  to  AC,  =250 

So  is  the  sine  of  A=46°  30' 


to  BC, 


=181.4 


As  radius  =90° 

is  to  AC,  =250 

So  is  the  sine  of  C=43°  30' 


toAB, 


=  172.1 


10.000000 
2.397940 
9.860562 

2.258502 

10.000000 
2.397940 

9.837812 

2.235752 


If  from  the  sum  of  the  second  and  third  logs,  that  of  the 
first  be  taken,  the  number  will  be  the  log.  of  the  fourth;  the 
number  answering  to  which  will  be  the  thing  required ;  but 
when  the  first  log.  is  radius,  or  10.000000,  reject  the  first  figure  of 
the  sum  of  the  other  two  logs,  (which  is  the  same  thing  as  to  sub- 
tract 10.000000),  and  that  will  be  the  log.  of  the  thing  required. 
2.  Making  AB  the  radius. 

Secant  A:  AC::  R:  AB. 
Secant  A':  AC  :  :  T.A  :  BC. 
That  is,  as  the  secant  of  A=46°  30'      10.162188 
is  to  AC,  =250  2.397940 

So  is  the  radius      =90°  10.000000 


to  AB,  =172.1 

As  the  secant  of  A    =46°  30' 

is  to  AC,  =250 

So  is  the  tangent  of  A =46°  30' 


to^C, 


=  181.34 


12.397940 

2.235762 

10.162188 

2.397940 

10.022750 

12.420690 
2.258502 


*  For  finding  the  logarithmic  sine,  co-sine,  &c.  of  any  number  of  de- 
grees and  minutes,  in  the  table,  also  the  degrees,  minutes,  &c.  of  any 
logarithmic  sine,  co-sine,  &c.,  the  reader  is  referred  to  table  2,  at  the  end 
of  this  treatise. 


TRIGONOMETRY. 


3.    Making  BC  the  radius. 
Sec.  C  :  AC 
Sec.  C  :  AC 
That  is,  as  the  secant  of  C 

is  to  AC, 
So  is  the  radius 


:  R  :  BC. 
:  T.CiAB. 

=43°  30'  10.139438 
=250  2.397940 
=90°  10.000000 


toBC,  =181.34 

As  the  secant  of  C     =43°  30' 

is  to  AC,  =250 

So  is  the  tangent  of  C=43°  30' 


12.397940 

2.258502 

10.139438 
2.397940 
9.977250 

12.375190 


to  AB,  =172.1  2.235752 

Or,  having  found  one  side,  the  other  may  be  obtained  by 
cor.  2,  theo.  14,  sect.  4. 

3d.    By  Gunter's  Scale. 

The  first  and  third  terms  in  the  foregoing  proportions  being 
of  a  like  nature,  and  those  of  the  second  and  fourth  being  also 
like  to  each  other ;  and  the  proportions  being  direct  ones ;  it 
follows,  that  if  the  third  term  be  greater  or  less  than  the  first, 
the  fourth  term  will  be  also  greater  or  less  than  the  second : 
therefore  the  extent  in  your  compasses  from  the  first  to  the 
third  term  will  reach  from  the  second  to  the  fourth. 

Thus,  to  extend  the  first  of  the  foregoing  proportions  ; 

1.  Extend  from  90°  to  46°  30',  on  the  line  of  sines;    that 
distance  will  reach  from  250,  on  the  line  of  numbers,  to  181, 
for  BC. 

2.  Extend  from  90°  to  '43°  30',  on  the  line  of  sines  ;  that 
distance  will  reach  from  250,  on  the  line  of  numbers,  to  172r 
for  A  B. 

If  the  first  extent  be  from  a  greater  to  a  less  number ;  when 
you  apply  one  point  of  the  compasses  to  the  second  term,  the 
other  must  be  turned  to  a  less  ;  and  the  contrary. 

By  def.  20,  sect.  4.  The  sine  of  90°  is  equal  to  the  radius  ?. 
and  the  tangent  of  45°  is  also  equal  to  the  radius  ;  because  if 
one  angle  of  a  right-angled  triangle  be  45°,  the  other  will  be 
also  45°  ;  and  thence  (by  the  lemma  preceding  theo.  7,  sect.  4) 
the  tangent  of  45°  is  equal  to  the  radius :  for  this  reason  the 
line  of  numbers  of  10.000000,  the  sine  of  90°,  and  tangent  of 
45°,  being  all  equal,  terminate  at  the  same  end  of  the  scale. 


TRIGONOMETRY.  89 

The  first  two  statings  of  this  case  answer  the  question 
without  a  secant ;  the  like  will  be  also  made  evident  in  all  the 
following  cases. 

4th.     Solution  by  Natural  Sines. 

From  the  foregoing  analogies,  or  statements,  it  is  obvious 
that  if  the  hypothenuse  be  multiplied  by  the  natural  sine  of 
either  of  the  acute  angles,  the  product  will  be  the  length  of  the 
side  opposite  to  that  angle ;  and  multiplied  by  the  natural  co- 
sine of  the  same  angle,  the  product  will  be  the  length  of  the 
other  side,  or  that  which  is  contiguous  to  the  angle.  Thus : 
The  given  angle =47°  30' 

Nat.  Sine=.725374  Nat.  Cos.=.688355 

Hyp.=        250  250 


36268700  34417750 

1450748  1376710 


Perpend.=  181.343500  Base=  172.088750 

CASE  II. 

The  base  and  angles  given,  to  find  the  perpendicular  and  hypothenuse. 


In  the  triangle  ABC,  there  is  the  angle  A  42°  20',  and  of 
course  the  angle  C  47°  40'  (by  cor.  2,  theo.  5),  and  the  side 
AB  190  given  ;  to  find  EC  and  AC. 

'  •  \ 
1st.    By  Construction. 

Make  the  angle  CAB  (by  prop.  16,  sect.  4)  in  blank  lines> 
as  before.  From  a  scale  of  equal  parts  lay  190  from  A  to  Bt 
on  the  point  B  erect  a  perpendicular  BC  (by  prob.  5,  sect.  4), 
the  point  where  this  cuts  the  other  blank  line  of  the  angle  will 
be  C  ;  so  is  the  triangle  ABC  constructed  :  let  AC  and  BC  be 
measured  from  the  same  scale  of  equal  parts  that  AB  was 
taken  from,  and  the  answers  are  found. 

2d.    By  Calculation. 

1.  Making  AC  the  radius. 

S.C-.ABi  :  R:AC. 
S.C  lABi:  S.A  :  BC. 


90  TRIGONOMETRY. 

That  is,  as  the  sine  of  C=47°  40' 

is  toAB,  =190 

So  is  radius  =90° 


to  AC,  =257 

As  the  sine  of  C         =47°  40' 

is  to  AB,         =190 
So  is  the  sine  of  A=42°  20' 


to  BC,  =173.1 

2.    Making  AB  the  radius. 

R:AB::T.A:  EC. 
R:AB::  Sec.  A  :  AC. 
That  is,  as  radius  =90° 

is  to  AB,  =190 

So  is  the  tangent  of  A— 42°  20' 


to  EC, 

As  radius 

is  to  AB,      =190 
So  is  the  secant  of  J.=42°  20 


=  173.1 
=90° 


to^tC,  =257 

3.    Making  BC  the  radius. 

T.C:  AB:  :  Sec.  C  :  AG 
T.C:AB::R:  BC. 

That  is,  as  the  tangent  of  C=47°  10' 

istoAB,  =190 

So  is  the  secant  of  C=47°  40' 


9.868785 

2.278754 

10.000000 

12.278754 

2.409969 

9.868785 
2.278754 
9.828301 

12.107055 
2.238270 


10.000000 
2.278754 
9.959516 

2.238270 

10.000000 

2.278754 
10.131215 

2.409969 


to  AC,      =257 

As  the  tangent  of  C=47°  40' 

isto^tf,  =190 

So  is  the  radius         =90° 


10.040484 

2.278754 

10.171699 

12.450453 

2.409969 

10.040484 

2.278754 

10.000000 

12.278754 


to  BC, 


=  173-1 


2.238270 


TRIGONOMETRY. 

Or,  having  found  one  of  the  required  sides,  the  other  maybe 
obtained  by  one  or  the  other  of  the  cors.  to  theo.  14,  sect.  4. 

3d.    By  Gunter's  Scale. 

1.  When  AC  is  made  the  radius. 

Extend  from  4?°  40'  to  90°  on  the  line  of  sines ;  that  dis- 
tance will  reach  from  190  to  257,  on  the  line  of  numbers, 
for  AC. 

2.  When  AB  is  made  the  radius,  the  first  stating  is  thus  per- 
formed : 

Extend  from  45°  on  the  tangents  (for  the  tangent  of  45°  i» 
equal  to  the  radius,  or  to  the  sine  of  90°  as  before)  to  42° 
20' ;  that  extent  will  reach  from  190,  on  the  line  of  numbers, 
to  173,  for  BC. 

3.  When  BC  is  made  the  radius,  the  second  stating  is  thus 
performed : 

Extend  from  47°  40',  on  the  line  of  tangents,  to  45°,  or  ra- 
dius ;  that  extent  will  reach  from  190  to  173,  on  the  line  of 
numbers,  for  BC;  for  the  tangent  of  47°  40'  is  more  than  the 
radius,  therefore  the  fourth  number  must  be  less  than  the  second, 
as  before. 

The  first  two  statings  of  this  case  answer  the  question 
without  a  secant. 

4th.     Solution  by  Natural  Sines. 


SofC.  SofC 

Nat.  S.  of  C.         Side  ABxR. 
Thus,  .739239)190.000000(257.02,  &c.=AC. 
147  8478 


4215220 
3696195 

5190250 
5174673 


1557700 

1478478 

and,  .673443 =Nat.  S  of 
190= side  AB. 


60609870 
673443 

127.954170 


92  TRIGONOMETRY. 

Nat.  Sof  C.739239)127.954170(173.09=.BC. 
73  9239 


5403027 
5174673 


2283540 
2217717 


6502300 
6653151 


CASE    III. 

The  angles  and  perpendicular  given,  to  find  the  base  and  hypothenuse. 

PL.  5.  fa.  6. 

In  the  triangle  ABC,  there  is  the  angle  A  40°,  and  conse- 
quently the  angle  C  50°,  with  BC  170,  given,  to  find  AC 
and  AB. 

1st.  By  Construction. 

Make  an  angle  CAB  of  40°  in  blank  lines  (by  prob.  16, 
sect.  4)  ;  with  BC  170  from  a  line  of  equal  parts  draw  the  lines 
EF  parallel  to  AB  (by  prob.  8,  sect.  4),  the  lower  line  of  the 
angle,  and  from  the  point  where  it  cuts  the  other  line  in  C  let 
fall  a  perpendicular  BC  (by  prob.  7,  sect.  4),  and  the  triangle 
is  constructed :  the  measures  of  A  C  and  AB,  from  the  same 
scale  that  BC  was  taken,  will  answer  the  question. 

What  lias  been  said  in  the  two  foregoing  cases  is  sufficient 
to  render  the  operations  in  this,  both  by  calculation,  Gunter's 
scale,  and  natural  sines,  so  obvious,  that  it  is  needless  to  insert 
them ;  however,  for  the  sake  of  the  learner,  we  give  for 
Answers,  AC  264.5,  and  AB  202.6. 


IV. 

The  base  and  hypothenuse  given,  to  find  the  angles  and  perpendicular. 

PL.  5.  fig.  7. 

In  the  triangle  ABC,  there  is  given  AB  300  and  AC  500;. 
the  angles  A  and  C  and  the  perpendicular  BC  are  required* 


TRIGONOMETRY. 
1st.    By  Construction. 


93 


From  a  scale  of  equal  parts  lay  300  from  A  to  B ;  on  B 
erect  an  indefinite  blank  perpendicular  line  ;  with  AC  500  from 
the  same  scale,  and  one  foot  of  the  compasses  in  A,  cross  the 
perpendicular  line  in  C ;  and  the  triangle  is  constructed. 

Byprob.  17,  sect.  4,  measure  the  angled,  and  let  BQ  be 
measured  from  the  same  scale  of  equal  parts  that  AC  and 
AB  were  taken  from ;  and  the  answers  are  obtained. 

2d.    By  Calculation. 

I. 'Making  AC  the  radius. 

AC:  R::AB  :  S.C. 
R:AC::S.A:BC. 

That  is,  as  AC 
is  to  radius, 
Sois^LB 


=500 
=90° 
=300 


2.698970 
10.000000 
2.477121 

12.477121 


to  the  sine  of  C,=  36°  52'  9.778151 

By  cor.  2,  theo.  5,  90°-— 36°  52'=53°  08',  the  angle  A. 

As  radius  =90°  10.000000 

isto.dC,  =500  2.698970 

So  is  the  sine  of  A    =53°  08'  9.903108 


to  BC, 

Making  AB  the  radius. 
AB:R 
R-.AB 
That  is,  as  AB 
is  to  radius, 
So  is  AC 


=400 


AC:  :  sec.  A. 
:  T.A  :  BC\ 

=300 
=90° 
=500 


to  the  secant  of  A,=53°  08' 

As  radius  =90° 

is  to  AB,  =300 

So  is  the  tangent  of  A= 5 3°  08' 


toBC, 


=400 


2.602078 


2.477121 

10.000000 

2.698970 

12.698970 

10.221849 

10.000000 

2.477121 

10.124990 

2.602111 


Or  BC  may  be  found  from  cor.  2,  theo.  14,  sect  4. 


94-  TRIGONOMETRY. 

3d.   By  Gunter's  Scale. 

1.  Making  AC  the  radius. 

Extend  from  500  to  300,  on  the  line  of  numbers ;  that  ex- 
tent will  reach  from  90°,  on  the  line  of  sines,  to  36°  52'  for 
the  angle  C. 

Again,  extend  from  90°  to  53°  08',  on  the  line  of  sines,  that 
extent  will  reach  from  500  to  400,  on  the  line  of  numbers, 
for  BC. 

2.  Making  AC  the  radius,  the  second  stating  is  thus  per- 
formed. 

Extend  from  radius,  or  the  tangent  of  45°,  to  53°  08',  that 
extent  will  reach  from  300  to  400,  for  BC. 

4th.    Solution  by  Natural  Sines.* 

RxAB                       .ACxSofA      nn 
=SofC;  and s =BC, 


AC  R 

Thus,  AC,         AB, 

5,00)300.0000,00 


.600000 =Nat.  sine  36°  52'. 

and, 
Nat.  sine  of  4=53°  8'=.800034 

AC  =  500 


400.017000=5C. 


CASE  V. 

The  perpendicular  and  hypothenuse  given,  to  find  the  angles  and  late. 

Pi.  5.  fig.  8. 

In  the  triangle  ABC  there  is  BC  306  and  AC  370  given, 
to  find  the  angles  A  and  C  and  the  base  AB. 

1st.    By  Construction. 

Draw  a  blank  line  from  any  point,  in  which  at  B  erect  a 
perpendicular,  on  which  lay  BC  306,  from  a  scale  of  equal 
parts:  from  the  same  scale,  with  A  C  370  in  the  compasses 

*  For  finding  the  natural  sines  and  co-sines,  the  reader  is  referred  to 
table  3. 


TRIGONOMETRY.  95 

draw  the  first  drawn  blank  line  in  A,  and  the  triangle  ABC  is 
constructed. 

Measure  the  angle  A  (by  prob.  17,  sect.  4),  and  also  AB, 
from  the  same  scale  of  equal  parts  the  other  sides  were  taken 
from,  and  the  answers  are  now  found. 

The  operations  by  calculation,  the  square  root,  Gunter's 
scale,  and  natural  sines  are  here  omitted,  as  they  have  been 
heretofore  fully  explained :  the  statings,  or  proportions,  must 
also  be  obvious,  from  what  has  already  been  said. 

Answers.  The  angle  A  55°  48' ;  therefore  the  angle  C34° 
12',  and  AB  208. 

CASE    VI. 

The  base  and  perpendicular  given,  to  find  the  angles  and  hypothenuse.^ 
PL.  5.  fig.  9. 

In  the  triangle  ABC,  there  is  AB  225  and  BC  272  given, 
to  find  the  angles  A  and  C  and  the  hypothenuse  AC. 

1st.    By  Construction. 

Draw  a  blank  line,  on  which  lay  AB  225,  from  a  scale  of 
equal  parts ;  at  B  erect  a  perpendicular ;  on  which  lay  BC 
272  from  the  same  scale ;  join  A  and  C,  and  the  triangle  is 
constructed. 

As  before,  let  the  angle  A  and  the  hypothenuse  AC  be 
measured,  in  order  to  find  the  answers. 

2d.    By  Calculation. 

1.  Making  A  B  the  radius. 

AB:  R::  BC:  T.A. 
R:AB:  :  sec.  A  :  AC. 

2.  Making  BC  the  radius. 

BC:  R::AB:  T.C. 
R:BC  :  :  sec.  C:  AC. 

By  calculation,  the  answers  from  the  foregoing  proportions 
are  easily  obtained  as  before. 

But  because  AC,  by  either  of  the  said  proportions,  is  found 
by  means  of  a  secant,  and  since  there  is  no  line  of  secants  on 
Gunter's  scale,  after  having  found  the  angles  as  before,  let  us 
suppose  AC  the  radius,  and  then 

l.S.A:BC::R:  AC 
or  2.  S.C-.AB:  :  R  :  AC. 


96  TRIGONOMETRY. 

These  proportions  may  be  easily  resolved,  either  by  calcula- 
tion or  Gunter's  scale,  as  before ;  and  thus  the  hypothenuse 
AC  may  be  found  without  a  secant. 

From  the  two  given  sides  the  hypothenuse  may  be  easily  ob- 
tained, from  cor.  1,  theo.  14,  sect. 

Thus,  the  square  of  AB= 50625 
Add  the  square  of  £C=73984 


65)346 
325 

703)2109 
2109 

From  what  has  been  said  on  logarithms,  it  is  plain, 

1.  That  half  the  logarithm  of  the  sum  of  the  squares  of  the 
two  sides  will  be  the  logarithm  of  the  hypothenuse.     Thus,* 

The  sum  of  squares,  as  before,  is  124609 ;  its  log.  is  5.095549, 
the  half  of  which  is  2.547774 ;  and  the  corresponding  number 
to  this  in  the  tables  will  be  353,  for  AC. 

2.  And  that  half  of  the  logarithm  of  the  difference  of  the 
squares  of  AC  and  AB,  or  of  AC  and  .BC,  will  be  the  loga- 
rithm of  EC,  or  of  ^5. 

The  following  examples  are  inserted  for  the  exercise  of  the 
learner. 

Ex.  1.  In  the  right-angled  triangle  AB C, 
n.         (  the  hypothenuse  AC  540  perches,  >  A        (  1?C300 
Given,}      ^33045,  JAns'UB449 

To  find  the  other  two  sides. 

Ex.  2.  In  the  right-angled  triangle  AB  C, 

p.        $  the  base  AB  162  chains,  <A  )  A        $  AC  270 
Given,^      33045,  jAns.   \BC^IQ 

To  find  the  other  two  sides. 

*  Demonstration.  The  square  of  the  hypothenuse  of  a  right-angled  tri- 
angle is  equal  to  the  sum  of  the  squares  of  the  sides  (theo.  14) ;  hence  the 
log.  of  (AC*ABZ)  =  the  log:  of  2JC2,  and  by  the  nature  of  logarithms  the 
log.  of  BC  is  equal  to  the  log.  of  BCZ  divided  by  2 ;  and  in  like  manner 
the  log.  of  (AB*+BC*)  =  the  log.  of  AC*,  hence  dividing  the  log.  of 
•  AC*  by  2  gives  the  log. of  AC.  Q,  E.  D. 


TRIGONOMETRY.  97 

Ex.  3.  In  the  right-angled  triangle  ABC, 
Sthe   perpendicular   BC    180  (  ,         $  AC   392.0146 
Glven'  I      links,  <C  62°  40'  \  ***  \  AB   348.2464 

To  find  the  other  two  sides. 

Ex.  4.  In  the  right-angled  triangle  ABC, 

n   $  the  hypothenuse  AC  392  poles,  >  .         ) 
Glven'  I      the  base  AS  180  poles  $  Ans*  ) 


° 


To  find  the  angles  and  perpendicular 

Ex.  5.  In  the  right-angled  triangle  ABC, 
the  hypothenuse  .AC  1198  )  r'<JL  54°  51 

chains,  the  perpendicular  >  Ans.  1  <  C  35°  09 
BC  980  chains  3  (  AB  690 

ic  angles  and  base. 

Ex.  6.  In  the  right-angled  triangle  ABC, 


n.  t  the  base  AR  735.9  links,  the  >  .  )  ^  ~  So  on> 
Given,  <  D/-I  ortrk  *  Ans.  <  <.A  23  3(X 

I  )      perpendicular  BC  320  $  -       /  Ar<   eno  * 

%  j  ••  111  I    _/L  \-/      c    F^.O 

To  find  the  angles  and  hypothenuse.  V 

OBLIQUE-ANGLED  PLANE  TRIGONOMETRY. 

BEFORE  we  proceed  to  the  solution  of  the  four  cases  of  Ob- 
lique-angled triangles,  it  is  necessary  to  premise  the  following 
theorems. 

THEOREM  I. 

PL.  5.  Jig.  10. 

1  In  any  plane  triangle  ABC  the  sides  are  proportional  to  the  sines  of  their 
opposite  angles ;  that  is,  S.C  :  AB  :  :  S.A  :  BC,  and  S.C  :  AB  :  :  S.B  : 
AC;  also,  S.B  :  AC  :  :  S.A  :  BC. 

By  theo.  10,  sect.  4,  the  half  of  each  side  is  the  sine  of  its 
opposite  angle;  but  the  sines *of  those  angles,  in  tabular  parts, 
are  proportional  to  the  sines  of  the  same  in  any  other  measure ; 
and  therefore  the  sines  of  the  angles  will  be  as  the  halves  of 
their  opposite  sides  ;  and  since  the  halves  are  as  the  wholes, 
it  follows  that  the  sines  of  their  angles  are  as  then*  opposite 
sides  ;  that  is,  S.C  :  AB  :  :  S.A  :  BC,  <fec.  Q.  E.  D. 

THEOREM  II. 

Fig.  11. 

In  any  plane  triangle  A  BC  the  sum  of  the  two  given  sides  AB  and  BC, 
including  a  given  angle  ABC,  is  to  their  difference  as  the  tangent  of  half 
the  sum  of  the  two  unknown  angles  A  and  C  is  ^  the  tangent  of  half  their 
difference. 

Produce  AB,  and  make  HB=BC,  and  join  HC :  let  fall  the 
E 


98  TRIGONOMETRY. 

perpendicular  BE,  and  that  will  bisect  the  angle  HBC  (by  theo. 
9,  sect.  4) ;  through  B  draw  BD  parallel  to  AC,  and  make 
HF=DC,  and  join  BF-,  takeBI=$A,  and  draw  1G  parallel 
to  BD  or  AC. 

It  is  then  plain  that  AH  will  be  the  sum  and  HI  the  differ- 
ence of  the  sides  AB  and  EC :  and  since  HB=BC,  and  BE 
perpendicular  to  HC,  therefore  HE=EC  (by  theo.  8,  sect.  4) ; 
and  since  BA—BI,  and  BD  and  IG  parallel  to  AC,  therefore 
GD=DC=FH,  and  consequently  HG=FD,  and  ±HG=iFD 
or  ED.  Again,  EBC,  being  half  J/J5C,  will  be  also  half 
the  sum  of  the  angles  A  and  C  (by  theo.  4,  sect.  4)  ;  also,  since 
HB,  HF,  and  the  included  angle  H  are  severally  equal  to  BC, 
CD,  and  the  included  angle  BCD.,  therefore  (by  theo.  6,  sect. 
4)  HBF=DBC=BCA  (by  part  2,  theo.  3,  sect.  4)  ;  and  since 
HBD=A  (by  part  3,  theo.  3,  sect.  4),  and  HBF=BCA, 
therefore  FBD  is  the  difference  and  EBD  half  the  difference 
of  the  angles  A  and  C  :  then  making  BE  the  radius,  it  is  plain 
that  EC  will  be  the  tangent  of  half  the  sum,  and  ED  the  tan- 
gent of  half  the  difference  of  the  two  unknown  angles  A  and 
C :  now  IG  being  parallel  to  AC,  AH  :  1H  :  :  CH  :  GH  (by 
cor.  1,  theo.  20,  sect.  4).  But  the  wholes  are  as  their  halves; 
that  is,  AH :  IH  :  :  CE  :  ED ;  that  is,  as  the  sum  of  the  two 
sides  AB  and  BC  is  to  their  difference,  so  is  the  tangent  of 
half  the  sum  of  the  two  unknown  angles  A  and  C  to  the  tan- 
gent of  half  their  difference.  Q.  E.  D. 

THEOREM  III. 

Fig.  12. 

In  any  right-lined  plane  triangle  ABD,  the  base  AD  will  be  to  the  sum  of 
the  other  sides  AB,  BD  as  the  difference  of  those  sides  is  to  the  difference  of 
the  segments  of  the  base  made  by  the  perpendicular  BE  ;  viz.  the  difference 
between  AE  and  ED. 

Produce  BD  ti\\BG=AB,  the  lesser  leg;  and  on  B  as  a 
centre,  with  the  distance  BG  or  BA,  describe  a  circle  AGHF, 
which  will  cut  BD  and  AD  in  the  points  H  and  F',  then  it  is 
plain  that  GD  will  be  the  sum,  and  HD  the  difference  of  the 
sides  AB  and  BD ;  also,  since  AE=EF  (by  theo.  8,  sect.  4), 
therefore  FDis  the  difference  of  AE,  ED,  the  segments  of  the 
base ;  but  (by  theo.  17,  sect.  4)  AD  :  GD  :  :  HD  :  FD ;  that 
is,  the  base  is  to  the  sum  of  the  other  sides  as  the  difference  of 
those  sides  is  to  the  differenced  the  segments  of  the  base. 
Q.  E.  D. 

Cor.  1.  In  the  above  triangle  the  longest  side  is  made  the 
base,  and  then  the  perpendicular  falls  within  the  triangle  ;  but 
if  DF  (the  same  construction  remaining  as  in  the  above,  only 


TRIGONOMETRY.  99 

joining  BF)  (fig.  3,  plate  14),  be  considered  the  base  of  the  tri- 
angle BDF,  then  BE  is  a  perpendicular  on  the  tyase  pro- 
duced ;  GD  is  equal  to  the  sum  of  the  sides  BF,  BD-,  HD  is 
equal  to  their  difference  :  also  AD  is  equal  to  the  sum  of  the 
segments  DE,  EF.  But  (by  theo.  17,  sect.  4)  FDxAD= 
GDxHD,  hence  FD  :  GD  :  :  HD  :  AD.  That  is,  as  the 
base  is  to  the  sum  of  the  two  sides,  so  is  the  difference  of  the 
sides  to  the  sum  of  the  segments  of  the  base.  Q.  E.  D. 

Cor.  2.  Hence  (by  calling  any  side  the  base)  as  the  base  is 
to  the  sum  of  the  sides,  so  is  the  difference  of  the  sides  to 
the  difference  or  sum  of  the  segments  of  the  base,  according  as 
the  perpendicular  Tails  within  or  without  the  triangle.* 

THEOREM  IV. 
Fig.  13. 

If  to  half  the  sum  of  two  quantities  be  added  half  their  difference,  the  sum 
vrill  be  the  greatest  of  them  ;  and  if  from  half  the  sum  be  subtracted  half 
their  difference,  the  remainder  will  be  the  least  of  them. 

Let  the  two  quantities  be  represented  by  AB  and  BC  (mak- 
ing one  continued  line),  whereof  AB  is  the  greatest,  and  BC 
the  least.  Bisect  the  whole  line  AC  in  E,  and  make  AD=BC; 
then  it  is  plain  that  AC  is  the  sum,  and  DB  the  difference  of 
the  two  quantities,  and  AE  or  EC  their  half-sum,  and  DE  or 
EB  their  half-difference.  Now  if  to  AE  we  add  EB,  we  shall 
have  AB  the  greatest  quantity ;  and  if  from  EC  we  take  EB, 
we  shall  have  BC  the  least  quantity.  Q.  E.  D. 

Cor.  Hence,  if  from  the  greatest  of  two  quantities  we  take 
half  the  difference  of  them,  the  remainder  will  be  half  their 
sum ;  or  if  to  half  their  difference  be  added  the  least  quantity, 
their  sum  will  be  half  the  sum  of  the  two  quantities. 

THEOREM  V. 

PL.  U.fig.  4. 

In  any  triangle  the  rectangle  under  two  sides  is  to  the  rec- 
tangle under  the  semiperimeter,  and  its  excess  above  the  base, 
as  the  square  of  the  radius  to  the  square  of  the  co-sine  of  half 
the  contained  angle. 

In  the  triangle  CBE,  the  perimeter  being  denoted  by  P,  CB 
X  CE  :  iP(±P—BE)  :  :  R3  :  cos.  ^C2.  Produce  EC  to  A, 

*  The  perpendicular  falls  within  or  without  the  triangle,  according  as 
the  square  of  the  greater  side  is  less  or  greater  than  the  sum  of  the 
squares  of  the  less  side  and  the  base.  For  a  demonstration  of  which  the 
reader  is  referred  to  (Prop.  12,  13,  B.  2)  Simpson's  Euclid. 

E  2 


100  TRIGONOMETRY. 

making  CA—CB-,  draw  BD  perpendicular  to  CE,  bisect  CE 
in  HI  and  join  AB. 

Let  OB  be  greater  than  EB,  then  (by  theo.  3,  fig.  12)  CE: 

C  7?2  ft  V^ 
CB+BE  :  :  CB—BE  :  —  ^  —  =2HD,by  adding  half  this 


--.!--  /~*  J7*  2 

to  half  the  base=  CH.  The  segment  CD—  --  —  -  -  --  ;  to 


u-     ^-    vt.       ™    •       ^n 

this  adding  (LI,  or  C#,  givesAD=-  --  9~CE 

_  (  CB+  CE)2  BE3  _CB+CE+BE  x  CB+CE—BE 

2CE  ~iCE^ 

Again,  AD=AC+CD=CB+CD;  hence  AD*  =  CB*+2 
CB.CD+CD*=2CB.AD;  also,  BD*  =  CB2CD2-,  hence 
=2.  C52  +2  CB.  CD  = 


=2CB.AD ;  therefore, 

\sjjj 

AB*=2CB.AD2,  or  P(±P—BE).AB2  =  CE.2CK.AD2,  di- 
viding both  sides  by  2  ;  CE.CB.AD2  =±P(\P— BE).AB2, 
consequently  CE.CB  :  ^P(^P — BE) :  :  AB*  :  AD2.  That 
is,  CExCB  :  ±Pxi(P—BE)  :  :  rad.3  :  (cos.  ±BCE)2.* 
Q.  E.  D. 

OBLIQUE-ANGLED  TRIANGLES. 

CASE    I. 

Two  sides  and  an  angle  opposite  to  one  of  them  given,  to  find  the  other 
angles  and  side. 

PL.  5.  fig.  14. 

In  the  triangle  ABC,  there  is  given  AB  240,  the  angle  A  46°  30',  and 
BC  200,  to  find  the  angle  C,  being  acute,  the  angle  B,  and  the  side  AC. 

1st.    By  Construction. 

Draw  a  blank  line,  on  which  set  AB  240,  from  a  scale  of 
equal  parts  ;  at  the  point  A,  of  the  line  AB,  make  an  angle  of  46° 
30',  by  an  indefinite  blank  line ;  with  BC  200,  from  a  like  scale 
of  equal  parts  that  AB  was  taken,  and  one  foot  in  B,  describe 
the  arc  DC  to  cut  the  last  blank  line  in  the  points  D  and  C. 
Now  if  the  angle  C  had  been  required  obtuse,  lines  from  D  to 
B,  and  to  A,  would  constitute  the  triangle  ;  but  as  it  is  re- 
quired acute,  draw  the  lines  from  C  to  B  and  to  A,  and  the 
triangle  ABC  is  constructed.  From  a  line  of  chords  let  the 
angles  B  and  C  be  measured;  and  AC  from  the  same  scale 

*  For  a  different  method  of  demonstrating  this  theorem,  as  well  as 
the  demonstration  of  other  useful  theorems,  the  reader  is  referred  to  Les- 
lie's Geometry  (pages  372,  373). 


TRIGONOMETRY.  101 

of  equal  parts  that  AB  and  BC  were  taken ;  E,nd.yoii.mli  have 
the  answers  required. 

2.     By  Calculation. 
This  is  performed  by  theo.  1  of  this  sect,  thus : 

As  BC  =200  2.301030 

is  to  the  sine  of  A,  =46°3a'  9.860562 

So  is  AS  =240  2.380211 


12.240773 


to  the  &ine  of  C,      =60°  31'  9.939743 

180°  —  the  sum  of  the  angles  A  and  C  will  give  the  angle 
;,  by  cor.  1,  theo.  5,  sect.  4. 
A  46°  30' 
C60    31 


180°— 107°  l'=72°  5&=B. 

As  the  sine  of  A=46°  30'  9.860562 

is  to  BC,         =200  2.301030 

So  is  the  sine  of  J?=72°  59'  9.980555 


12.281585 
to  AC,  =-263.  7  2.421023 

3d.    By  Gunter's  Scale. 

Extend  from  200  to  240  on  the  line  of  numbers ;  that  dis- 
tance will  reach  from  46°  30',  on  the  line  of  sines,  to  60°  31', 
for  the  angle  C. 

Extend  from  46°  30'  to  72°  59',  on  the  line  of  sines ;  that 
distance  will  reach  from  200  to  263.7  on  the  line  of  numbers, 
for  AC. 

Note. — The  method  by  natural  shies  will  be  obvious  from 
the  foregoing  analogies. 

If  the  side  opposite  the  given  angle  be  equal  to  or  greater 
than  the  other  given  side,  or  the  given  angle  obtuse,  then  there 
would  be  but  one  answer  to  the  problem,  because  the  angle 
opposite  that  other  given  side  will  be  always  acute  ;  but  when 
the  given  angle  is  acute,  and  opposite  the  less  of  the  given  sides, 
the  answer  is  ambiguous,  as  the  sine  of  an  angle  is  equal  to  the 
sine  of  its  supplement,  consequently  the  required  angle  oppo- 
site that  other  given  side  may  be  obtuse  or  acute,  unless  it  is 
given  in  the  conditions  of  the  problem. 


TRIGONOMETRY. 

~  In-the  last'  problem  the  given  angle  is  acute,  and  the  side  op- 
posite to  it  less  than  the  other  given  side,  therefore  the  angle 
C  may  be  acute  or  obtuse ;  but  the  side  and  angle  answering 
to  the  acute  value  of  C  has  been  already  found.  Now  it  re- 
mains to  find  the  side  and  angle  of  the  triangle  answering  to 
the  obtuse  value  of  C,  which  is  thus  found  : 

The  acute  value  of  C,  found  in  the  foregoing  calculation,  is 
60°  31',  consequently  its  obtuse  value  is  180°— 60°  31'=  119° 
29';  then  119°  29'+46°  30',  taken  from  180°,  gives  14°  l'=| 
to  the  remaining  angle  ABC  (pi.  14,  fig.  5). 

To  find  the  side  AC,  answering  to  the  obtuse  value  of  the 
angle  C. 

As  the  sine  of  A  =46°  30'         9.860562 


is  to  EC,  =200  2.301030 

So  is  sine  B          =14°  1'  9.384182 


11.685212  f  « 

•( 

to  the  side  AC,  =66.8  1.824650 

CASE  II. 

Two  angles  and  a  side  given,  to  find  the  other  sides. 
PL.  5.  fig.  15. 

In  the  triangle  ABC  there  is  the  angle  A  46°  30',  AB  230,  and  the  angle 
B  37°  30'  given,  to  find  AC  and.  BC. 

1st.    By  Construction. 

Draw  a  blank  line,  upon  which  set  AB  230  from  a  scale  of 
equal  parts ;  at  the  point  A  of  the  line  AB  make  an  angle  of 
46°  30',  by  a  blank  line  ;  and  at  the  point  B  of  the  line  AB 
make  an  angle  of  37°  30',  by  another  blank  line ;  the  intersec- 
tion of  those  lines  gives  the  point  C :  then  the  triangle  ABC 
is  constructed.  Measure  AC  and  BC  from  the  same  scale  of 
equal  parts  that  AB  was  taken,  and  you  have  the  answer  re- 
quired. 

2d.    By  Calculation. 
By  cor.  "1,  theo.  5,  sect  4,  180°  —  the  sum  of  the  angles  .4. 

A  46°  30' 

537    30 



180°— 84°  O0'=96°  00'=C. 


TRIGONOMETRY.  103 

By  def.  27,  sect.  4.     The  sine  of  96°  =  the  sine  of  84°, 
which  is  the  supplement  thereof;  therefore,  instead  of  the  sine 
of  96°,  look  in  the  tables  for  the  sine  of  84°. 
By  theo.  1  of  this  sect. 

As  the  sine  of  C      =96°  00'       9.997614 

is  to  AB,  =230  2.361728 

So  is  the  sine  of  A  =46°  30'       9.860562 

12.222290 


to  BC,  =167.8          2.224676 

As  the  sine  of  C      =96°  00'       9.997614 

is  to  AB,  =230  2.361728 

So  is  the  sine  of  A  =37°  30'       9.784447 

12.146175 


to  AC,  =140.8          2.148561 

3d.    By  Gunter's  Scale. 

Extend  from  84°  (which  is  the  supplement  of  96°)  to  46° 
30'  on  the  sines  ;  that  distance  will  reach  from  230  to  168,  on 
the  line  of  numbers,  for  BC. 

Extend  from  84°  to  37°  30',  on  the  sines ;  that  extent  will 
reach  from  230  to  141,  on  the  line  of  numbers,  for  AC. 
t 

CASE    III. 

Two  sides  and  a  contained  angle  given,  to  find  the  other  angles  and  side. 
PL.  5.  Jig.  16. 

In  the  triangle  ABC,  there  is  AB  240,  the  angle  A  36°  40',  and  AC 
180  given,  to  find  the  angles  C  and  B  and  the  side  BC. 

1st.    By  Construction. 

Draw  a  blank  line,  on  which,  from  a  scale  of  equal  parts, 
lay  AB  240  ;  at  the  point  A  of  the  line  AB  make  an  angle 
of  36°  40',  by  a  blank  line  ;  on  which  from  A  lay  AC  180, 
from  the  same  scale  of  equal  parts ;  measure  the  angles  C 
and  B  and  the  side  BC,  as  before,  and  you  have  the  answers 
required. 

2d.    By  Calculation. 

By  cor.  1,  theo.  5,  sect.  4,  180^  — the  angle  A  36°  40'=»- 
143°  20',  the  sum  of  the  angles  C  and  B :  therefore,  half  of 


104  TRIGONOMETRY. 

143°  20'  will  be  half  the  sum  of  the  two  required  angles  C 
and  B. 

* 

By  theo.  2  of  this  sect. 

As  the  sum  of  the  two  sides  AB  and  AC=42Q 
is  to  their  difference,  =  60 

So  is  the  tangent  of  half  the  sum  of  )      =71°  40' 
the  two  unknown  angles  C  and  B  ) 
to  the  tangent  of  half  their  difference, =23°  20' 

• 
By  theo.  4. 

To  half  the  sum  of  the  angles  C  and  5=71°  40' 

Add  half  their  difference   as  now  found— 23°  20' 

The  sum  is  the  greatest  angle,  or  ang.  C=95  00 
Subtract,  and  you  have  the  least  angle,  or  5— 48  20 

The  angles  C  and  B  being  found,  BC  is  had  as  before,  by 
theo.  1  of  this  sect.  Thus, 

S.B  :AC::S.A:  BC. 
48°  20'  :  180  :  :  36°  40'  :  143.9. 

3d.     By  Gunter's  Scale. 

Because  the  two  first  terms  are  of  the  same  kind,  extend 
from  420  to  60  on  the  line  of  numbers;  lay  that  extent 
from  45°  on  %>  i|n»  of  tangents,  and  keeping  the  left  leg  of 
your  compasses  fixed,  move  the  right  leg  to  71°  40';  that  dis- 
tance laid  from  45°  on  the  same  line  will  reach  to  23°  30',  the 
half-difference  of  the  required  angles.  Whence  the  angles  are 
obtained,  as  before. 

The  second  proportion  may  be  easily  extended,  from  what 
has  been  already  said. 

.\, 

CASE  IV. 

PL.  5.  fig.  17. 

The  three  sides  given,  to  find  the  angles. 

In  the  triangle  ABC,  there  is  given  AB  64,  4C47,  J3C34;  the  angles 
A,  B,  and  C  are  required. 

1st.    By  Construction. 

The  construction  of  this  triangle  must  be  manifest,  from 
prob.  1,  sect,  4, 


TRIGONOMETRY.  105 

2d.    By  Calculation. 

From  the  point  C  let  fall  the  perpendicular  CD  on  the  base 
AB,  and  it  will  divide  the  triangle  into  two  right-angled  ones, 
ADC  and  CBD,  as  well  as  the  base  AB  into  the  two  seg- 
nents  AD  and  DB. 

AC    47 
EC    34 

Sum   81 
Difference  13 

By  theo.  3  of  this  sect. 

As  the  base  or  the  longest  side  AB  64 

is  to  the  sum  of  the  other  sides  AC  and  JBC,  81 

So  is  the  difference  of  those  sides  13 

to  the  difference  of  the  segments  of  the  base  AD,  DJ5, 16.46 

By  theo.  4  of  this  sect. 

To  half  the  base,  or  to  half  the  sum  of  the  segments  \  Q0 

AD  and  DB, .  $  d 

Add  half  their  difference,  now  found,  8.23 

Their  sum  will  be  the  greatest  segment  AD,  40.23 

Subtract,  and  their  difference  will  be  the  least  seg- 
ment,DB, 

In  the  right-angled  triangle  ADC,  there  is  AC  47  and  AD 
40.23  given,  to  find  the  angle  A. 

This  is  resolved  by  case  4  of  right-angled  plane  trigonome- 
try, thus  : 

AD  :  R  :  :  AC  :  sec.  A 
40.23  :  90°  :  :  47  :  31°  08' 

Or  it  may  be  had  by  finding  the  angle  ACD,  the  complement 
of  the  angle  A,  without  a  secant,  thus : 

AC  :  R  :  :  AD  :  S.ACD 
44  :  90°  :  :  40.23  :  58°  52' 
90°_58°  52'=31°  08',  the  angle  A. 
Then  by  theo.  1  of  this  sect. 

BC  :  S.A  :  :  AC  :  S.B 
-••-  34  :  31°  08'  :  :  47  :  45°  37' 

E3 


106  TRIGONOMETRY. 

By  cor.  l,theo.  5,  sect.  4,  180°  —  the  sum  of  A  and  B=C. 
'  JL31°  08' 


B45   37 


180°—  76   45=103°  15',  the  angle  C. 

3d.     By  Gunter*s  Scale. 

The  first  proportion  is  extended  on  the  line  of  numbers  ;  and 
it  is  no  matter  whether  you  extend  from  the  first  to  the  third  or 
to  the  second  term,  since  they  are  all  of  the  same  kind  :  if  you 
extend  to  the  second,  that  distance  applied  to  the  third  will  give 
the  fourth  ;  but  if  you  extend  from  the  first  to  the  third,  that 
extent  will  reach  from  the  second  to  the  fourth.* 

The  methods  of  extending  the  other  proportions  have  been 
already  fully  treated  of. 

RULE  2. 

Either  of  the  angles,  -as  A,  may  be  found  by  adding  together 
the  arithmetical  complements  of  the  logarithms  of  the  two  sides 
AB,  AC,  containing  the  required  angle,  the  log.  of  the  half- 
sum  of  the  three  sides,  and  the  log.  of  the  difference  between 
the  half-sum  and  the  side  opposite  the  required  angle  ;  then 
half  the  sum  of  these  four  logarithms  will  bf  the  logarithmic 
co-sine  of  half  the  required  angle,  f  It  is  required  to  find  the 
angle  A,  in  the  last  problem,  by  this  rule,  the  sides  remaining 
the  same. 


AC=47  Api  Co.  7.327902 

§±  Ar.  Co.  7.193820 


2)145 
Half-sum        72.5'  Log.  2.860338 


Difference  38.51  Log.  2.585461 


2)19.967521 

Cos.  i  A,  15°  34'         9.983760 
Whose  double  31°  08'  is  the  angle  A. 

*  The  reader  is  referred  to  Hutton's  Mathematics,  vol.  ii.  New- York 
edition,  for  the  method  of  investigating  Plane  Trigonometry  analytically. 

t  The  demonstration  of  this  rule  is  evident  from  theo.  5,  and  the  nature 
of  logarithms ;  but  in  working  the  proportion  by  logarithms,  we  omit 


TRIGONOMETRY.  107 

If  the  other  angles  were  required,  they  can  be  found  by  Case 
1,  or  by  theo.  1  of  this  sect. 

RULE  3.* 

Add  the  three  sides  together,  and  take  half  the  sum  and  the 
differences  between  the  half-sum  and  each  side :  then  add  the 
complements  of  the  logarithms  of  the  half-sum  and  of  the  dif- 
ference between  the  half-sum  and  the  side  opposite  to  the  angle 
sought,  to  the  logarithms  of  the  differences  of  the  half-sum  and 
the  other  sides :  half  their  sum  will  be  the  tangent  of  the  angle 
required. 

Example.  In  the  triangle  ABC,  having  the  side  AB  562, 
AC  800,  and  BC  320,  to  find  the  angle  ABC. 


AC=80Q 


Sum  1682 


#=841  Ar.  Co.  7.075204 

H—AC=4l  Ar.  Co.  8.387216 

H—AB=279  log.  2.445604 

H—BC=52l  log.  2.716838 


Sum  20.624862 

i- 


Tang,  of  64°  2'— £  sum  10.312431 

Whose  double  128°  4'  is  the  angle  AB  C.     Whence 
other  angles  can  be  easily  found  by  theo.  1  of  this  section. 

An  example  in  each  case  of  oblique-angled  triangles. 

1.  In  the  triangle  ABC,  having  AB  106,  AC  65,  and  the 
angle  B  31°  49',  to  find  the  La  A  and  C  and  the  side  BC. 

Ans.     The  L.  C=59°  17'  or  120°  43',  the  LA  27°  28'  or 
88°  54',  and  the  side  BC=43.2  or  123.2. 

2.  In  the  triangle  ABC,  having  the  sideAB  2200,  the  L  A 
35°,  and  the  L  B  47°  24',  to  find  the  sides  AC  and  BC  and 
the  L  C. 

Ans.     The  LC  97°  36',  the  side  AC  1636,  and  the  side 
BC  1272. 

3.  In  the  triangle  ABC,  having   the  side  AB  240,  AC 

the  log.  of  the  square  of  radius  or  20,  which  is  just  equivalent  to  re- 
jecting 20  from  the  sum  of  the  four  logarithms,  which  should  be  done, 
because  for  every  arithmetical  complement  that  is  taken  10  must  be  re- 
jected :  but  the  Ar.  Co.  of  the  two  sides  containing  the  required  angle  is 
taken  ;  consequently  20  should  be  rejected,  which  is  equal  to  the  log.  of 
the  square  of  radius. 

*  For  the  demonstration  of  this  rule  the  reader  is  referred  to  Leslie's 
Geometry,  prop.  12,  p.  372. 


108  TRIGONOMETRY. 

263.7,  and  the  angle  A  46°  30',  to  find  the  other  angles  and  the 
side  BC. 

Ans.     The  L.C  60°  31',  the  L.B   72°  59',  and  the  side 
BC  200. 

4.    In  the  triangle  ABC,  having  the  sides  given,  viz.  AB= 

144.8,  BC=W9,  and  AC=76,  it  is  required  to  find  the  angles 
by  each  of  the  three  rules  given  to  Case  4. 

Ans.     The  least  angle  29°  49',  next  greater  54°  07',  and  the 
greatest  96°  04'. 

Additional  exercises,  with  their  answers. 

QUESTIONS    FOR    EXERCISE. 

1.  Given  the  hypothenuse  108,  and  the  angle  opposite  the 
perpendicular  25°  36' ;  required  the  base  and  perpendicular. 

Ans.     The  base  is  97.4,  and  the  perpendicular  46.66. 

2.  Given  the  base  96,  and  its  opposite  angle  71°  45' ;  re- 
quired the  perpendicular  and  the  hypothenuse. 

Ans.     The  perpendicular  is  31.66,  and  the  hypothenuse 
101.1. 

3.  Given  the  perpendicular  360,  and  its  opposite  angle  58° 
20' ;  required  the  base  and  the  hypothenuse. 

Ans.     The  base  is  222,  and  the  hypothenuse  423. 

4.  Given  the  base  720,  and  the  hypothenuse  980  ;  required 
the  angles  and  the  perpendicular. 

Ans.     The  angles  are  47°  17'  and  42°  43',  and  the  perpen- 
dicular 664.8. 

5.  Given  the   perpendicular   110.3,  and  the  hypothenuse 
176.5  ;  required  the  angles  and  the  base. 

Ans.     The  angles  are  38°  41'  and  51°  19',  and  the  base 
137.8.       , 

6.  Given  the  base  360,   and  the  perpendicular  480;  re- 
quired the  angles  and  the  hypothenuse. 

Ans.     The  angles  are  53°  8'  and  36°  52',  and  the  hypothe- 
nuse 600. 

7.  Given  one  side  129,  an  adjacent  angle  56°  30',  and  the 
opposite  angle  81°  36' ;  required  the  third  angle  and  the  remain- 
ing sides. 

Ans.     The  third  angle  is  41°  54',  and  the  remaining  sides 
are  108.7  and  87.08. 

8.  Given  one  side  96.5,  another  side  59.7,  and  the  angle 
opposite  the  latter  side  31°  30' ;  required  the  remaining  angles 
and  the  third  side. 

Ans.     This  question  is  ambiguous,  the  given  side  opposite  the 
given  angle  being  less  than  the  other  given  side  (see  Rule  1); 


OF  THE  CHAIN.  109 

hence,  if  the  angle  opposite  the  side  96.5  be  acute,  it  will 
be  57°  3«',  the  remaining  angle  90°  52',  and  the  third  side 
114.2;  but  if  the  angle  opposite  the  side  96.5  be  obtuse,  it 
will  be  122°  22',  the  remaining  angle  26°  8',  and  the  third  side 
50.32. 

9.  Given   one  side  110,  another  side  102,  and  the  con- 
tained angle  113°  36';  required  the  remaining  angles  and  the 
third  side. 

Ans.  The  remaining  angles  are  34°  37'  and  31°  47',  and 
the  third  side  is  177.5. 

10.  Given  the  three  sides  respectively  120.6,  125.5,  and 
146.7  ;  required  the  angles. 

Ans.     The  angles  are  51°  53',  54°  58',  and  73°  9'. 

The  student  who  has  advanced  thus  far  in  this  work  with 
diligence  and  active  curiosity  is  now  prepared  to  study,  with 
ease  and  pleasure,  the  following  Part,  which  comprehends  all 
the  necessary  directions  for  the  practice  of  Surveying. 


PART   II. 

THE  PRACTICAL  SURVEYOR'S  GUIDE. 
SECTION  I. 

Containing  a  particular  Description  of  the  several  Instruments  used  t» 
Surveying^  with  their  respective  Uses. 

THE  CHAIN. 

THE  stationary  distance,  or  merings  of  ground,  are  measured 
either  by  Gunter's  chain  of  four  poles  or  perches,  which  con- 
sists of  100  links  (and  this  is  the  most  natural  division),  or  by 
one  of  50  links,  which  contains  two  poles  or  perches  :  but  be- 
cause the  length  of  a  perch  differs  in  many  places,  therefore 
the  length  of  chains  and  their  respective  links  will  differ  also. 

The  English  statute-perch  is  5|  yards,  the  two-pole  chain  is 
1 1  yards,  and  the  four-pole  one  is  22  yards  ;  hence  the  length 
of  a  link  in  a  statute-chain  is  7.92  inches. 

For  the  more  ready  reckoning  the  links  of  a  four-pole  chain, 
there  is  a  large  ring,  or  sometimes  a  round  piece  of  brass,  fixed 
at  every  10  links  ;  and  at  50  links,  or  in  the  middle,  there  are 


110  OF  THE  CHAIN. 

two  large  rings.  In  such  chains  as  have  a  brass  piece  at  every 
10  links,  there  is  the  figure  1  on  the  first  piece,  2  on  the  second, 
8  on  the  third,  &c.  to  9.  By  leading  therefore  that  end  of  the 
chain  forward  which  has  the  least  number  next  to  it,  he  who 
carries  the  hinder  end  may  easily  determine  any  number  of 
links :  thus,  if  he  has  the  brass  piece  number  8  next  to  him, 
and  six  links  more  in  a  distance,  that  distance  is  86  links. 
After  the  same  manner  10  may  be  counted  for  every  large  ring 
of  a  chain  which  has  not  brass  pieces  on  it ;  and  the  number  of 
links  is  thus  readily  determined. 

The  two-pole  chain  has  a  large  ring  at  every  10  links,  and 
in  its  middle, or  at  25  links, there  are  two  large  rings;  so  that  any 
number  of  links  may  be  the  more  readily  counted  off,  as  before. 

The  surveyor  should  be  careful  to  have  his  chain  measured 
before  he  proceeds  on  business  ;  for  the  rings  are  apt  to  open  by 
frequently  using  it,  and  its  length  is  thereby  increased,  so  that 
no  one  can  be  too  circumspect  in  this  point. 

In  measuring  a  stationary  distance,  there  is  an  object  fixed 
in  the  extreme  point  of  the  line  to  be  measured ;  this  is  a  di- 
rection for  the  hinder  chainman  to  govern  the  foremost  one  by, 
in  order  that  the  distance  may  be  measured  in  a  right  line ;  for 
if  the  hinder  chainman  causes  the  other  to  cover  the  object,  it 
is  plain  the  foremost  is  then  in  a  right  line  towards  it.  For 
this  reason  it  is  necessary  to  have  a  person  that  can  be  relied 
on  at  the  hinder  end  of  the  chain,  in  order  to  keep  the  foremost 
man  in  a  right  line ;  and  a  surveyor  who  has  no  such  person 
should  chain  himself.  The  inaccuracies  of  most  surveys  arise 
from  bad  chaining,  that  is,  from  straying  out  of  the  right  line, 
as  well  as  from  other  omissions  of  the  hinder  chainman :  no 
person,  therefore,  should  be  admitted  at  the  hinder  end  of  the 
chain  of  whose  abilities,  in  this  respect,  the  surveyor  is  not 
previously  convinced  ;  since  the  success  of  the  survey,  in  a  great 
measure,  depends  on  his  care  and  skill. 

In  setting  out  to  measure  any  stationary  distance,  the  fore 
man  of  the  chain  carries  with  him  ten  iron  pegs  pointed,  each 
about  ten  inches  long ;  and  when  he  has  stretched  the  chain  to 
its  full  length,  he  at  the  extremity  thereof  sticks  one  of  those 
pegs  perpendicularly  in  the  ground ;  and  leaving  it  there,  he 
draws  on  the  chain  till  the  hinder  man  checks  him  when  he 
arrives  at  that  peg  :  the  chain  being  again  stretched,  the  fore 
man  sticks  down  another  peg,  and  the  hind  man  takes  up  the 
former ;  and  thus  they  proceed  at  every  chain's  length  con- 
tained in  the  line  to  be  measured,  counting  the  surplus  links 
contained  between  the  last  peg  and  the  object  at  the  terminar 
tion  of  the  line,  as  before :  so  that  the  number  of  pegs  taken 


OF  THE  CHAIN.  Ill 

up  by  the  hinder  chainman  expresses  the  number  of  chains  : 
to  which,  if  the  odd  links  be  annexed,  the  distance  line  required 
in  chains  and  links  is  obtained,  which  must  be  registered  in  the 
field-book,  as  will  hereafter  be  shown. 

If  the  distance  exceeds  10,  20,  30,  <fcc.  chains,  when  the 
leader's  pegs  are  all  exhausted,  the  hinder  chainman,  at  the 
extremity  of  the  10  chains,  delivers  him  all  the  pegs ;  from 
whence  they  proceed  to  measure  as  before,  till  the  leader's  pegs- 
are  again  exhausted,  and  the  hinder  chainman  at  the  extremity 
of  these  10  chains  again  delivers  him  the  pegs,  from  whence 
they  proceed  to  measure  the  whole  distance  line  in  the  like 
manner ;  then  it  is  plain,  that  the  number  of  pegs  the  hinder 
chainman  has  being  added  to  10,  if  he  had  delivered  all  the 
pegs  once  to  the  leader,  or  to  20  if  twice,  or  to  30  if  thrice,  &c., 
will  give  the  number  of  chains  in  that  distance ;  to  which  if 
the  surplus  links  be  added,  the  length  of  the  stationary  distance 
is  known  in  chains  and  links. 

It  is  customary,  and  indeed  necessary,  to  have  red,  or  other 
coloured  cloth  fixed  to  the  top  of  each  peg,  that  the  hinder  man 
at  the  chain  may  the  more  readily  find  them ;  otherwise,  in. 
chaining  through  corn,  high  grass,  briers,  rushes,  &c.  it  would 
be  extremely  difficult  to  find  the  pegs  which  the  leader  puts 
down :  by  this  means  no  time  is  lost,  which  otherwise  must 
be,  if  no  cloths  are  fixed  to  the  pegs,  as  before. 

It  will  be  necessary  here  to  observe,  that  all  slant,  or  inclined 
surfaces,  as  sides  of  hills,  are  measured  horizontally,  and  not 
on  the  plane  or  surface  of  the  hill,  and  is  thus  effected. 


PL.  8.  fig.  4. 

Let  ABC  be  a  hill ;  the  hindmost  chainman  is  to  hold  the 
end  of  the  chain  perpendicularly  over  the  point  A  (which  he 
can  the  better  effect  with  a  plummet  and  line,  than  by  letting  a 
stone  drop,  which  is  most  usual),  as  d  is  over  ^4,  while  the  leader 
puts  down  his  peg  at  e :  the  eye  can  direct  the  horizontal  position 
near  enough ;  but  if  greater  accuracy  were  required,  a  quadrant 
applied  to  the  chain  would  settle  that.  In-  the  same  manner 
the  rest  may  be  chained  up  and  down ;  but  in  going  down,  it  is 
plain  the  leader  of  the  chain  must  hold  up  the  end  thereof,  and 
the  plummet  thence  suspended  will  mark  the  point  where  he  is 
to  stick  his  peg.  The  figure  is  sufficient  to  render  the  whole 
evident,  and  to  show  that  the  sum  of  the  chains  will  be  the 
horizontal  measure  of  the  base  of  the  hill :  for  de=Ao,fg= 
op,  hi=pq,  &c. ;  therefore  de+fg+ki,  &,c.—Ao+op+pq,  &c. 


112  OF  THE  CHAIN. 


,*  the  base  of  the  hill.  If  a  whole  chain  cannot  be  car- 
ried horizontally,  half  a  chain,  or  less,  may,  and  the  sura  of 
these  half-chains,  or  links,  will  give  the  base,  as  before. 

If  the  inclined  side  of  the  hill  be  the  plane  surface,  the  angle 
of  the  hill's  inclination  may  be  taken,  and  the  slant  height  may 
be  measured  on  the  surface  ;  and  thence  (by  Case  1  of  right- 
angled  trigonometry)  the  horizontal  line  answering  to  the  top 
may  be  found  ;  and  if  we  have  the  angle  of  inclination  given 
on  the  other  side,  with  those  already  given,  we  can  find  the 
horizontal  distance  across  the  hill,  by  Case  2  of  oblique  trigo- 
nometry. 

All  inclined  surfaces  are  considered  as  horizontal  ones  ;  for 
all  trees  which  grow  upon  any  inclined  surface  do  not  grow 
perpendicular  thereto,  but  to  the  plane  of  the  horizon  :  thus,  if 
Ad,  ef,  gh,  &c.  were  trees  on  the  side  of  a  hill,  they  grow  per- 
pendicular to  the  horizontal  base  AC,  and  not  to  the  surface 
AB  :  hence  the  base  will  be  capable  to  contain  as  many  trees 
as  are  on  the  surface  of  the  hill,  which  is  manifest  from  the 
continuation  of  them  thereto.  And  this  is  the  reason  that  the 
area  of  the  base  of  a  hill  is  considered  to  be  equal  in  value 
to  the  hill  itself. 

Besides,  the  irregularities  of  the  surfaces  of  hills  in  general 
are  such,  that  they  would  be  found  impossible  to  be  determined 
by  the  most  able  mathematicians.  Certain  regular  curve  sur- 
faces have  been  investigated,  with  no  small  pains,  by  the  most 

*  The  number  of  chains  taken  down  in  the  field-book  is  longer  than 
the  lines  Ao,  op,  pq,  &c.,  because  the  chain,  being  elevated  above  the  sur- 
face of  the  earth  (though  stretched  with  a  force  at  both  ends),  forms  a 
curve,  which  approaches  a  right  line,  according  as  the  force  is  more  or  less 
applied  ;  but  does  not  coincide  with  it  :  as,  for  example.  —  Let  the  chain  be 
stretched  from  d  to  e  (PI.  8.  fig.  4)  ;  it  does  not  coincide  with  de,  but  forms 
a  curve  line,  which  must  be  longer  than  de  or  its  equal  Ao,  and  so  isfg 
or  op  shorter  than  the  chain,  and  in  like  manner  with  all  the  rest.  And 
de,fg,  &c.=Ao,  op,  &,c.=AG  ;  consequently,  the  number  of  chains,  being 
greater  than  de,  ef,  &c.  or  Ao,  op,  &c.  is  greater  than  A  C  ;  therefore,  the 
horizontal  line  AC  (by  surveyors  in  general)  is  made  too  long,  therefore 
a  deduction  must  be  made  for  every  chain  in  the  field-book  ;  tho  sum  to 
be  taken  from  A  C  may  be  found  by  making  an  experiment  on  a  two-pole 
chain  (when  extended  above  the  surface  of  the  earth  by  a  force  at  both 
extremities),  and  measuring  the  distance  from  its  middle  point  to  the 
middle  of  the  right  line  which  would  join  its  extremities,  which  call  a,  and 
call  ^  the  length  of  the  chain  b  ;  then  ^/J2  —  a*=$de,  or  half  the  right 
line  ;  therefore  2^b2—a2—de,  or  the  right  line;  from  whence  2i  — 
2^/i2  —  a2=  the  excess  for  every  chain  which  is  measured  or  taken  down 
in  the  field-book  :  calling  the  number  of  chains  c,  then  c.2b  —  2v/(i-  —  a-)= 
the  whole  excess  on  the  honzontal  line  AC.  From  what  is  here  demon- 
strated, the  practitioner  will  be  able  to  find  the  sum  to  be  taken  from 
every  horizontal  line  in  surveying  hills,  &c. 


OF  THE  CHAIN.  113 

eminent ;  therefore  an  attempt  to  determine  in  general  the  in- 
finity of  irregular  surfaces  which  offer  themselves  to  our  view, 
to  any  degree  of  certainty,  would  be  idle  and  ridiculous,  and 
for  this  reason  also,  the  horizontal  area  is  only  attempted. 

Again,  if  the  circumjacent  lands  of  a  hill  be  planned  or 
mapped,  it  is  evident  we  shall  have  a  plan  of  the  hill's  base  in  the 
middle :  but  were  it  possible  to  put  the  hill's  surface  in  lieu 
thereof,  it  would  extend  itself  into  the  circumjacent  lands,  and 
render  the  whole  a  heap  of  confusion :  so  that  if  the  surfaces 
of  hills  could  be  determined,  no  more  than  the  base  could  be 
mapped. 

Roads  are  usually  measured  by  a  wheel  for  that  purpose, 
called  the  perambulator,  to  which  there  is  fixed  a  machine,  at 
the  end  whereof  there  is  a  spring,  which  is  struck  by  a  peg  in 
the  wheel  once  in  every  rotation ;  by  this  means  the  number 
of  rotations  is  known ;  if  such  a  wheel  were  3  feet  4  inches 
diameter,  one  rotation  would  be  K)i  feet,  which  is  half  a  plan- 
tation perch;  and  because  320  perches  make  a  mile,  therefore 
640  rotations  will  be  a  mile  also  ;  and  the  machinery  is  so  con- 
trived that  by  means  of  a  hand,  which  is  carried  round  by  the 
work,  it  points  out  the  miles,  quarters,  and  perches,  or  some- 
times the  miles,  furlongs,  and  perches. 

Or  roads  may  be  measured  by  a  chain  more  accurately; 
for  80  four-pole,  or  160  two-pole  chains,  or  320  perches,  make 
a  mile  as  before  :  and  if  roads  are  measured  by  a  statute  chain, 
it  will  give  you  the  miles  English,  but  if  by  a  plantation  chain,  the 
miles  will  be  Irish.  Hence  an  English  mile  contains  1760, 
and  an  Irish  mile  2240  yards ;  and  because  14  half-yards  is 
an  Irish,  and  11  half-yards  is  an  English  perch,  therefore  11 
Irish  perches,  or  Irish  miles,  are  equal  to  14  English  ones. 

Since  some  surveys  are  taken  by  a  four-pole  and  others  by 
a  two-pole  chain,  and  as  ground  for  houses  is  measured  by  feet, 
we  will  show  how  to  reduce  one  to  the  other  in  the  following 
problems. 


PROBLEM  I. 

To  reduce  two-pole  chains  and  links  to  four-pole  ones. 

If  the  number  of  chains  be  even,  the  half  of  them  will  be  the 
four-pole  ones,  to  which  annex  the  given  links.     Thus : 

1.    In  16cA.  37J.  of  two-pole  chains,  how  many  four-pole 
ones  1  Answer  8cA.  37/. 

But  if  the  number  of  chains  be  odd,  take  the  half  of  them  for 


114  OF  THE  CHAIN. 

chains,  and  add  50  to  the  links,  and  they  will  be  four-pole 
chains  and  links.     Thus : 

2.    In  17 ch.  421  of  two-pole  chains  how  many  four-pole 
ones  ?  Answer  Sch.  921. 


PROBLEM  II. 

To  reduce  four-pole  chains  and  links  to  two-pole  ones. 

Double  the  chains,  to  which  annex  the  links  if  they  be  less 
than  50 ;  but  if  they  exceed  50  double  the  chains,  add  one  to 
them,  and  take  50  from  the  links,  and  the  remainder  will  be  the 
links.     Thus : 
ch.    I. 

1.    In  8.  37  of  four-pole  chains  how  many  two-pole  ones  ? 
2 


16.  37 

ch.   I 

2.     In  8.  82  of  four-pole  chains  how  many  two-pole  ones  1 
2.  50 

17.  32,  Answer. 

PROBLEM  III. 

To  reduce  four-pole  chains  and  links  to  perches  and  decimals  of  a  perch. 

The  links  of  a  four-pole  chain  are  decimal  parts  of  it,  each 
link  being  the  hundredth  part  of  a  chain ;  therefore  if  the 
chain  and  links  be  multiplied  by  4  (for  4  perches  are  a  chain), 
the  product  will  be  the  perches  and  decimal  parts  of  a  perch. 
Thus: 

ch.    I 

How  many  perches  in  13.  64  of  four-pole  chains? 

4 

Answer,  54.  56  perches. 

PROBLEM  IV. 

To  reduce  two-pole  chains  and  links  to  perches  and  decimals  of  a  perch. 

They  may  be  reduced  to  four-pole  ones  (by  prob.  1),  and 
thence  to  perches  and  decimals  (by  the  last)  ;  or, 

If  the  links  be  multiplied  by  4,  carrying  one  to  the  chains  when 
the  links  are,  or  exceed,  25 ;  and  the  chains  by  2,  adding  1  if 


OF  THE  CHAIN.  115 

occasion  be ;  the  product  will  be  the  perches  and  decimals  of 
a  perch.     Thus : 

eh.  I. 

1.    In  17.  21  of  two-pole  chains  how  many  perches? 
2.    4 

« 
Answer,  34.  84  perches. 

ch.  L 

2.   In  15.  38  of  two-pole  chains  how  many  perches? 
2.     4 

Answer,  31.  52  perches. 

PROBLEM  V. 

To  reduce  perches  and  decimals  of  a  perch  to  four-pole  chains  and  links. 

Divide  by  4,  so  as  to  have  two  decimal  places  in  the  quo- 
tient, and  that  will  be  four-pole  chains  and  links.     Thus : 
In  31.52  perches  how  many  four-pole  chains  and  links? 

ch.    I 
4)31.52(7.  88,  Answer. 


35 
32 


PROBLEM  VI. 

To  reduce  perches  and  decimals  of  a  perch  to  two-pole  chains  and  links. 

The  perches  may  be  reduced  to  four-pole  chains  (by  the  last), 
and  from  thence  to  two-pole  chains  (by  prob.  2) ;  or, 

Divide  the  whole  number  by  2,  the  quotient  will  be  chains ; 
to  the  remainder  annex  the  given  decimals,  and  divide  by  4 ; 
the  last  quotient  will  be  the  links.  Thus  : 

In  31.52  perches  how  many  two-pole  chains  and  links? 

ch.    L 
2)31.52(15.  38,  Answer. 

11 


4)152(38 
32 


116  OF  THE  CHAIN. 

PROBLEM  VII. 

To  reduce  chains  and  links  to  feet  and,  decimal  parts  of  a  foot. 

If  they  be  two-pole  chains,  reduce  them  to  four-pole  ones 
(by  prob.  1) :  these  being  multiplied  by  the  feet  in  a  four-pole 
chain  will  give  the  feet  and  decimals  of  a  foot.  Thus : 

In  I7ch.  21/.  of  two-pole  chains  how  many  feet? 
ch.   I. 

8.  71  of  four-pole  chains. 
66  feet  =  1  chain. 


5226 
5226     Answer,  574ft.  I0±in. 


Feet  574.86 
12 


Inches  10.32 
4 

1.28 

PROBLEM  VIII. 

To  reduce  feet  and  inches  to  chains  and  links. 

Reduce  the  inches  to  the  decimal  of  a  foot,  and  annex  that  to 
the  feet ;  that  divided  by  the  feet  in  a  four-pole  chain  will  give 
the  four-pole  chains  and  links  in  the  quotient ;  these  may  be 
reduced  to  two-pole  chains  and  links,  if  required,  by  prob.  2. 
Thus : 

In  217/Z.  9in.  how  many  two-pole  chains? 
12)9.00(.75  the  decimal  of  9  inches. 

60 

66)217.75(3.29  of  four-pole  chains,  or  6cA.  29/. 
197 
655 
61 


OF  THE  CHAIN.  117 

How  to  take  a  survey  by  the  CHAIN  only. 
PROBLEM  I. 

To  survey  a  piece  of  ground,  by  going  round  it,  and  the  method,  of  taking 
the  angles  of  the  field,  by  the  chain  only. 

PL.  6.^.6. 

Let  ABCDEFG  be  a  piece  of  ground  to  be  surveyed:  be- 
ginning at  the  point  A,  let  one  chain  be  laid  in  a  direct  line  from 
A  towards  G,  where  let  a  peg  be  left,  as  at  c ;  and  again  the 
like  distance  from  A  in  a  direct  line  towards  B,  where  another 
peg  is  also  to  be  left,  as  at  d ;  let  the  distance  from  d  to  c  be 
measured,  and  placed  in  the  field-book  in  the  second  column 
under  the  denomination  of  angles,  in  a  line  with  station  No.  1 ; 
and  in  the  same  line,  under  the  title  of  distances  in  the  third 
column,  let  the  measure  of  the  line  AB  in  chains  and  links  be 
inserted.  Being  now  arrived  at  B,  let  one  chain  be  laid  in  a 
direct  line  from  B  towards  A,  where  let  a  peg  be  left,  as  at/, 
and  again  the  like  distance  from  B  in  a  direct  line  towards  C, 
where  let  also  another  peg  be  left,  as  at  e ;  the  distance  from  e 
tofis  to  be  inserted  in  the  field-book,  in  the  second  column, 
under  angles,  in  a  line  with  station  No.  2 ;  and  in  the  same 
line,  under  the  title  of  distances  in  the  third  column,  let  the 
measure  of  the  line  UC,in  chains  and  links,  be  inserted  :  after 
the  same  manner  we  may  proceed  from  C  to  D,  and  thence  to 
JB;  but  because  the  angle  at  E,  viz.  FED,  is  an  external 
angle,  after  having  laid  one  chain  from  E  to  h,  and  to  g,  the 
distance  from  g  to  h  is  measured  and  inserted  in  the  column 
of  angles,  in  a  line  with  station  No.  5,  and  on  the  side  of  the 
field-book  against  that  station  we  make  an  asterisk,  thus  *,  or 
any  other  mark,  to  signify  that  to  be  an  external  angle,  or  one 
measured  out  of  the  ground.  Proceed  we  then  as  before  from 
E  to  F,  to  G.  and  thence  to  A,  measuring  the  angles  and  dis- 
tances, and  placing  them  as  before  in  the  field-book  opposite 
to  their  respective  stations  :  so  will  the  field-book  be  completed 
in  the  manner  following. 

N.  B. — After  this  manner  the  angles  for  inaccessible  distances 
may  be  taken,  and  the  method  of  constructing  or  laying  them 
down,  as  well  as  the  construction  of  the  map,  from  the  follow- 
ing field-notes,  must  be  obvious  from  the  method  of  taking  them. 

The  form  of  the  field-book,  with  the  title. 

A  Field-Book  of  part  of  the  land  of  Grange,  in  the  parish  of 
Portmarnock,  barony  of  Coolock,  and  county  of  Dublin ; 
being  part  of  the  estate  of  L.  P.,  Esq.,  let  to  C.  D-.,  fanner. 
Surveyed  January  30,  1782. 


118 


OF  THE  CHAIN, 

Taken  by  a  four-pole  chain. 


Remarks. 

No. 
Sta. 

Angles. 
eh.  I 

Distance. 
ch.     L 

Mr.  J.  D.'s  part  of  Grange  .  .  . 

Mr.  L.  P.  'a  part  of  Portmarnock  > 

Strand                                         $ 
* 

Widow  J.  G.'s  part  of  Grange    . 

1 
2 
3 
4 
5 
6 
7 

.  80 
.  79 
.  76 
.  41i 
'.  87^ 
.  14 
.  89 

17.  65 
18.  50 
28.  00 
20.  00 
14.  83 
19.  41 
24.  53 

Close  at  the  first  station. 

Explanation  of  the  Remarks. 

Mr.  J.  D's  part  of  Grange  bounds  or  is  adjacent  to  the  sur- 
veyed land  from  the  first  to  the  third  station  ;  Mr.  L.  P's  part 
of  Portmarnock  bounds  it  from  the  third  to  the  fourth  station  ; 
the  strand  then  is  the  boundary  from  thence  to  the  sixth  ;  and 
from  the  sixth  to  the  first  station,  the  widow  J.  G's  part  of 
Orange  is  the  boundary. 

It  is  absolutely  necessary  to  insert  the  persons'  names,  and 
town-lands,  strands,  rivers,  bogs,  rivulets,  &c.  which  bound  or 
circumscribe  the  land  which  is  surveyed,  for  these  must  be  ex- 
pressed in  the  map. 

In  a  survey  of  a  town-land,  or  estate,  it  is  sufficient  to  men- 
tion only  the  circumjacent  town-lands,  without  the  occupiers' 
names  :  but  when  a  part  only  of  a  town-land  is  surveyed,  then 
it  is  necessary  to  insert  the  person  or  persons'  names  who  hold 
any  particular  parcel  or  parcels  of  such  town-land  as  bound  the 
part  surveyed. 

When  an  angle  is  very  obtuse,  as  most  in  our  present  figure 
are,  viz.  the  angles  .  at  A,  B,  C,  E,  and  6?,  it  will  be  best  to 
lay  a  chain  from  the  angular  point,  as  at  J.,  on  each  of  the 
containing  sides  to  c  and  to  d  ;  and  any  where  nearly  in  the 
middle  of  the  angle,  as  at  e  :  measuring  the  distances  ce  and 
ed  ;  and  these  may  be  placed  for  the  angle  in  the  field-book. 
Thus, 

No.  Sta.     Angle. 

ch.    L  ch.     L 

1.  03 
1.09 


,- 
17'65 


For  when  an  angle  is  very  obtuse,  the  chord  line,  as  cd,  will  be 
nearly  equal  to  the  radii  Ac  and  Ad  ;  so  if  the  arc  ced  be  swept, 
and  the  chord  line  cd  be  laid  on  it,  it  will  be  difficult  to  deter- 


OF  THE  CHAIN. 


119 


mine  exactly  that  point  in  the  arc  where  cd  cuts  it :  but  if  the 
angle  be  taken  in  two  parts,  as  ce  and  ed,  the  arc,  and  the  angle 
thence,  may  be  truly  determined  and  constructed. 

After  the  same  manner  any  piece  of  ground  may  be  surveyed 
by  a  two-pole  chain. 

PROBLEM  II. 

To  take  a  survey  of  apiece  of  ground  from  any  point  within  it,  from 
whence  all  the  angles  can  be  seen,  by  the  chain  only. 

PL.  6.  fig.  6. 

Let  a  mark  be  fixed  at  any  point  in  the  ground,  as  at  H,  from 
whence  all  the  angles  can  be  seen ;  let  the  measures  of  the 
lines  HA,  HB,  HC,  &c.  be  taken  to  every  angle  of  the  field 
from  the  point  H ;  and  let  those  be  placed  opposite  to  No.  1, 
2,  3,  4,  &c.  in  the  second  column  of  the  radii :  the  measures  of 
the  respective  lines  of  the  mering,  viz.  AB,  BC,  CD,  DE,  &c. 
being  placed  in  the  third  column  of  distances,  will  complete  the 
field-book.  Thus : 


Remarks. 

No. 

Radii. 
ch.    I. 

Distance. 
ch.    I 

ki%C 

20.  00 

17.  65 

2 

21.  72 

18.  50 

3 

21.  74 

28.  00 

4 

25.  34 

20.  00 

5 

17.  20 

14.  83 

6 

29.  62 

19.  41 

7 

21.  20 

24.  53 

Close  at  the  first  station. 

If  any  line  of  the  field  be  inaccessible,  as  suppose  CD  to  be, 
then  by  way  of  proof  that  the  distance  CD  is  true,  let  the  mea- 
sure of  the  angle  CHD  be  taken  by  the  line  00,  with  the  chain : 
if  this  angle  corresponds  with  its  containing  sides,  the  length 
of  the  line  DC  is  truly  obtained,  and  the  whole  work  is  truly 
taken. 

Note. — That  hi  setting  off  an  angle,  it  is  necessary  to  use 
the  largest  scale  of  equal  parts,  viz.  that  of  the  inch,  which  is 
diagonally  divided  into  100  parts,  in  order  that  the  angle  should 
be  accurately  laid  down ;  or  if  two  inches  were  thus  divided 
for  angles,  it  would  be  the  more  exact ;  for  it  is  by  no  means 
necessary  that  the  angles  should  be-  laid  from  the  said  scale 
.with  the  stationary  distances. 


120  OF  THE  CHAIN. 

PROBLEM  III. 

To  take  a  survey  by  the  chain  onlyy  when  all  the  angles  cannot  be  seen  from 
one  point  within. 

PL.  6.  fig.  7. 

Let  the  ground  to  be  surveyed  be  represented  by  1,  2,  3,  4, 
&c.  Since  all  the  angles  cannot  be  seen  from  one  point,  let  us 
assume  three  points,  as  A,  #,  C,  from  whence  they  may  be  seen ; 
at  each  of  which  let  a  mark  be  put,  and  the  respective  sides 
of  the  triangle  be  measured  and  set  down  in  the  field-book ; 
let  the  distance  from  A  to  1,  and  from  B  to  1,  be  measured, 
and  these  will  determine  the  point  1 ;  let  the  other  lines 
which  flow  from  A,  J5,  C,  as  well  as  the  circuit  of  the  ground, 
be  then  measured  as  the  figure  directs ;  and  thence  the  map 
may  be  easily  constructed.  i 

There  are  other  methods  which  may  be  used  ;  as  dividing 
the  ground  into  triangles,  and  measuring  the  three  sides  of  each; 
or  by  measuring  the  base  and  perpendicular  of  each  triangle. 
But  this  we  shall  speak  of  hereafter. 

PROBLEM  IV. 

How  to  take  any  inaccessible  distance  by  the  chain  only. 
PL.  8.  fig.  8. 

Suppose  AB  to  be  the  breadth  of  a  river,  or  any  other  inac- 
cessible distance,  which  may  be  required. 

Let  a  staff  or  any  other  object  be  set  at  J5,  draw  yourself 
backward  to  any  convenient  distance  C,  so  that  B  may  cover 
A  ;  from  U,  lay  off  any  other  distance  by  the  river's  side  to  E, 
and  complete  the  parallelogram  EBCD  :  stand  at  D,  and  cause 
a  mark  to  be  set  at  JF,  in  the  direction  of  A  ;  measure  the 
distance  in  links  from  E  to  JP,  and  FB  will  be  also  given. 
Wherefore  EF  :  ED  :  :  FB  :  AB.  Since  it  is  plain  (from  part 
1,  theo.  3,  sect.  4,  and  theo.  2,  sect.  4)  the  triangles  JEjPDand 
BFA  are  mutually  equiangular. 

If  part  of  the  chain  be  drawn  from  B  to  C,  and  the  other 
part  from  B  to  E  ;  and  if  the  ends  at  E  and  C  be  kept  fast,  it 
will  be  easy  to  turn  the  chain  over  to  Z>,  so  as  to  complete  a 
parallelogram ;  by  reckoning  off  the  same  number  of  links 
you  had  in  BC,  from  E  to  D,  and  pulling  each  part  straight. 


THE  CIRCUMFERENTOR.        121 


THE  CIRCUMFERENTOR. 

THIS  instrument  is  composed  of  a  brass  circular  box,  about 
five  or  six  inches  in  diameter ;  within  which  is  a  brass  ring, 
divided  on  the  top  into  360  degrees,  and  numbered  10,  20,  30, 
&c.  to  360 :  in  the  centre  of  the  box  is  fixed  a  steel  pin  finely 
pointed,  called  a  centre-pin,  on  which  is  placed  a  needle 
touched  by  a  loadstone,  which  always  retains  the  same  situa- 
tion ;  that  is,  it  always  points  to  the  north  and  south  points  of 
the  horizon  nearly,  when  the  instrument  is  horizontal,  and  the 
needle  at  rest. 

The  box  is  covered  with  a  glass  lid  in  a  brass  rim,  to  pre- 
vent the  needle  being  disturbed  by  wind  or  rain  at  the  time  of 
surveying:  there  is  also  a  brass  lid  or  cover,  which  is  laid 
over  the  former  to  preserve  the  glass  in  carrying  the  instrument. 

This  box  is  fixed  by  screws  to  a  brass  index  or  ruler  of  about 
14  or  15  inches  in  length,  to  the  ends  whereof  are  fixed  brass 
sights  which  are  screwed  to  the  index  and  stand  perpendicular 
thereto :  in  each  sight  is  a  large  and  a  small  aperture  or  slit, 
one  over  the  other;  but' these  are  changed,  that  is,  if  the  large 
aperture  be  uppermost  in  the  one  sight,  it  will  be  lowest  in  the 
other,  and  so  of  the  small  ones :  therefore  the  small  aperture 
in  one  is  opposite  to  the  large  one  in  the  other,  in  the  middle  of 
which  last  there  is  placed  a  horse-hair  or  fine  silk  thread. 

The  instrument  is  then  fixed  on  a  ball  and  socket,  by  the  help 
of  which  and  a  screw  you  can  readily  fix  it  horizontally  in  any 
given  direction,  the  socket  being  fixed  on  the  head  of  a  three- 
legged  staff,  whose  legs,  when  extended,  support  the  instrument 
while  it  is  used. 

To  take  field-notes  by  the  Circumferentor. 
PL.  6.  fig.  6. 

Let  your  instrument  be  fixed  at  any  angle  as  A,  your  first 
station ;  and  let  a  person  stand  at  the  next  angle  J5,  or  cause  a 
staff  with  a  white  sheet  to  be  set  there  perpendicularly  for  an 
object  to  take  your  view  to :  then  having  placed  your  instru- 
ment horizontally  (which  is  easily  done  by  turning  the  box  so 
that  the  ends  of  the  needle  may  be  equidistant  from  its  bottom, 
and  it  traverses  or  plays  freely)  turn  the  flower-de-luce,  or 
north  part  of  the  box,  to  your  eye,  and  looking  through  the 
small  aperture  turn  the  index  about  till  you  cut  the  person  or 
object  in  the  next  angle  B  with  the  horse-hair  or  thread  of  the 
opposite  sight ;  the  degrees  then  cut  by  the  south  end  of  the 

F 


122  THE  CIRCUMFERENTOR, 

needle  will  give  the  number  to  be  placed  in  the  second  column 
of  your  field-book  in  a  line  with  station  No.  1,  and  expresses 
the  number  of  degrees  the  stationary  line  is  from  the  north, 
counting  quite  round  with  the  sun. 

|  Most  needles  are  pointed  at  the  south  end,  and  have  a  small 
ring  at  the  north :  such  needles  are  better  than  those  which  are 
pointed  at  each  end,  because  the  surveyor  cannot  mistake  by 
counting  to  a  wrong  end,  which  error  may  be  frequently  com- 
mitted in  using  a  two-pointed  needle. 

•  Two-pointed  needles  have  sometimes  a  ring,  but  more  usually 
a  cross  towards  the  north  end ;  and  the  south  end  is  generally 
bearded  towards  its  extremity,  and  sometimes  not,  but  its  arm 
is  a  naked  right  line  from  the  cap  at  the  centre. 

Having  taken  the  degrees  or  bearing  of  the  first  stationary 
line  AB,  let  the  line  be  measured,  and  the  length  thereof  in 
chains  and  links  be  inserted  in  the  third  column  of  your  field- 
book,  under  the  title  of  distances,  opposite  to  station  No.  1. 

It  is  customary,  and  even  necessary,  to  cause  a  sod  to  be 
dug  up  at  each  station  or  place  where  you  fix  the  instrument, 
to  the  end  that  if  any  error  should  arise  in  the  field-book  it  may 
be  the  more  readily  adjusted  and  corrected,  by  trying  over  the 
former  bearings  and  stationary  distances. 

Having  done  with  your  first  station,  set  the  instrument  over 
the  hole  or  spot  where  your  object  stood,  as  at  J5,  for  your 
second  station,  and  send  him  forward  to  the  next  angle  of  the 
field,  as  at  C ;  and  having  placed  the  instrument  in  a  horizon- 
tal direction,  with  the  sights  directed  to  the  object  at  C,  and  the 
north  of  the  box  next  your  eye,  count  your  degrees  to  the  south 
end  of  the  needle,  which  register  in  your  field-book  in  the  sec- 
ond column  opposite  to  station  No.  2 ;  then  measure  the  sta- 
tionary distance  BC,  which  insert  in  the  third  column ;  and 
thus  proceed  from  angle  to  angle,  sending  your  object  before 
you,  till  you  return  to  the  place  where  you  began,  and  you  will 
have  the  field-book  complete ;  observing  always  to  signify  the 
parties'  names  who  hold  the  contiguous  lands,  and  the  names 
of  the  town-lands,  rivers,  roads,  swamps,  lakes,  &c.  that  bound 
the  land  you  survey,  as  before ;  and  this  is  the  manner  of  taking 
field-notes  by  what  is  called  fore-sights. 

But  the  generality  of  mearsmen  frequently  set  themselves  in 
disadvantageous  places,  so  as  often  to  occasion  two  or  more 
stations  to  be  made  where  one  may  do,  which  creates  much 
trouble  and  loss  of  time  ;  we  will  therefore  show  how  this  may 
be  remedied,  by  taking  back-sights,  thus :  let  your  object  stand 
at  the  point  where  you  begin  your  survey,  as  at  A ;  leaving 
him  there,  proceed  to  your  next  angle  JB,  where  fix  your  instru- 


THE  CIRCUMFERENTOR.  123 

nrent  so  that  you  may  have  the  longest  view  possible  towards 
C.  Having  set  the  instrument  in  a  horizontal  position,  turn  the 
south  part  of  the  box  next  your  eye,  and  having  cut  your  object 
at  A,  reckon  the  degrees  to  the  south  point  of  the  needle,  which 
will  be  the  same  as  if  they  were  taken  from  the  object  to  the 
instrument,  the  direction  of  the  index  being  the  same.  Let  the 
degree  be  inserted  in  the  field-book,  and  the  stationary  dis- 
tance be  measured  and  annexed  thereto  in  its  proper  column ; 
and  thus  proceed  from  station  to  station,  leaving  your  object  in 
the  last  point  you  left  till  you  return  to  the  first  station  A. 

By  this  method  your  stations  are  laid  out  to  the  best  advan- 
tage, and  two  men  may  do  the  business  of  three,  for  one  of 
those  who  chain  may  be  your  object ;  but  in  fort-sights  you 
must  have  an  object  before  you,  besides  two  chainmen. 

It  was  said  before,  that  a  surveyor  should  have  a  person  with 
him  to  carry  the  hinder  end  of  the  chain,  on  whom  he  can  de- 
pend :  this  person  should  be  expert  and  ready  at  taking  offsets, 
as  well  as  exact  in  giving  a  faithful  return  of  the  length  of  every 
stationary  line.  One  who  has  such  a  person,  and  who  uses 
back-sights,  will  be  able  to  go  over  nearly  double  the  ground 
he  could  in  the  same  time  by  taking  fore-sights,  because  of 
overseeing  the  chaining;  for  should  he  take  back-sights  he 
must  be  obliged,  after  taking  his  degree,  to  go  back  to  the  fore- 
going station,  to  oversee  the  chaining,  and  by  this  means  to 
walk  three  times  over  every  line,  which  is  a  labour  not  to  be 
borne. 

Or  a  back  and  a  fore-sight  may  be  taken  at  one  station,  thus  : 
with  the  south  of  the  box  to  your  eye,  observe  from  B  the  ob- 
ject Aj  and  set  down  the  degree  in  your  field-book  cut  by  the 
south  end  of  the  needle.  Again,  from  B  observe  an  object  at 
C,  with  the  north  of  the  box  to  your  eye,  and  set  down  the  de- 
gree cut  by  the  south  point  of  the  needle,  so  have  you  the  bear- 
ings of  the  lines  AB  and  BC ;  you  may  then  set  up  your  in- 
strument at  D,  from  whence  take  a  back-sight  to  C  and  a  fore- 
sight to  E :  thus  the  bearings  may  be  taken  quite  round,  and 
the  stationary  distances  being  annexed  to  them  will  complete 
the  field-book. 

But  in  this  last  method  care  must  be  taken  to  see  that  the 
sights  have  not  the  least  cast  on  either  side ;  if  they  have,  it 
will  destroy  all :  and  yet  with  the  same  sights  you  may  take  a 
survey  by.  fore-sights,  or  by  back-sights  only,  with  as  great  truth 
as  if  the  sights  were  ever  so  erect,  provided  the  same  cast  con- 
tinues without  any  alteration  ;  but,  upon  the  whole,  back-sights 
only  will  be  found  the  readiest  method 

If  your  needle  be  pointed  at  each  end,  in  taking  fore-sights 
F2 


124  THE  CIRCUMFERENTOR. 

you  may  turn  the  north  part  of  the  box  to  your  eye,  and  count 
your  degrees  to  the  south  part  of  the  needle,  as  before ;  or  you 
may  turn  the  south  of  the  box  to  your  eye,  and  count  your  de- 
grees to  the  north  end  of  the  needle. 

But  in  back-sights  you  may  turn  the  north  of  the  box  to  your 
eye,  and  count  your  degrees  to  the  north  point  of  the  needle ; 
or  you  may  turn  the  south  of  the  box  to  your  eye,  and  count 
your  degrees  to  the  south  end  of  the  needle. 

The  brass  ring  in  the  box  is  divided  on  the  side  into  360  de- 
grees, thus :  from  the  north  to  the  east  into  90,  from  the  north 
to  the  west  into  90,  from  the  south  to  the  east  into  90,  and  from 
the  south  to  the  west  into  90  degrees  ;  so  the  degrees  are  num- 
bered from  the  north  to  the  east  or  west,  and  from  the  south  to 
the  east  or  west. 

The  manner  of  using  this  part  of  the  instrument  is  this :  hav- 
ing directed  your  sights  to  the  object,  whether  fore  or  back,  as 
before,  observe  the  two  cardinal  points  of  your  compass  the 
point  of  the  needle  lies  between  (the  north,  south,  east,  and 
west  being  called  the  four  cardinal  points,  and  are  graved  on 
the  bottom  of  the  box),  putting  down  those  points  together  by 
their  initial  letters,  and  thereto  annexing  the  number  of  degrees, 
counting  from  the  north  or  south,  as  before,  thus  ;  if  the  point  of 
your  needle  lies  between  the  north  and  east,  north  and  west,  south 
and  east,  or  south  and  west  points  in  the  bottom  of  the  box, 
then  put  down  NE,  NW1  SE,  or  S  W,  annexing  thereto  the 
number  of  degrees  cut  by  the  needle  on  the  side  of  the  ring, 
counting  from  the  north  or  south,  as  before. 

But  if  the  needle  point  exactly  to  the  north,  south,  east,  or 
west,  you  are  then  to  write  down  N,  S,  £,  or  TF,  without  an- 
nexing any  degree. 

This  is  the  manner  of  taking  field-notes,  whereby  the  con- 
tent of  ground  may  be  universally  determined  by  calculation ; 
and  they  are  said  to  be  taken  by  the  quartered  compass  or  by 
the  four  nineties. 


To  find  the  number  of  degrees  contained  in  any  given  angle. 

Set  up  your  instrument  at  the  angular  point,  and  thence  di- 
rect the  sights  along  each  leg  of  the  angle,  and  note  down  their 
respective  bearings,  as  before  ;  the  difference  of  these  bearings, 
if  less  than  180,  will  be  the  quantity  of  degrees  contained  in  the 
given  angle ;  but  if  more  take  it  from  360,  and  the  remainder 
will  be  the  degrees  contained  in  the  given  angle. 

Ex.  Let  the  angle  proposed  be  GAB  (pi.  6,  fig.  6) ;  place 
the  instrument  at  A,  with  the  flower-de-luce  towards  you ;  then 


THE  THEODOLITE.  125 

direct  the  sights  to  .B,  and  observe  what  degrees  are  cut  by  the 
south  end  of  the  needle,  which  let  be  250° ;  then  turning  the 
instrument  about  on  its  stand,  direct  the  sights  to  G,  note  again 
what  degrees  are  cut  by  the  south  end  of  the  needle,  which  sup- 
pose are  172°.  Then  250°  —  1 72°  =  68°  =  the  L  GAB ;  but 
if  the  degrees  cut  should  be  298°  and  105°,  then  298° — 105° 
=  193°,  which  taken  from  360°  leaves  167°  =  the  L.  GAB. 


THE  THEODOLITE. 

Fig.  1.     Frontispiece. 

THIS  instrument  is  a  circle,  commonly  of  brass,  of  ten  or 
twelve  inches  in  diameter,  whose  limb  is  divided  into  360  de- 
grees, and  those  again  are  subdivided  into  smaller  parts,  as  the 
magnitude  of  it  will  admit ;  sometimes  by  equal  divisions  and 
sometimes  by  diagonals  drawn  from  one  concentric  circle  of  the 
limb  to  another. 

In  the  middle  is  fixed  a  circumferentor  with  a  needle;  but 
this  is  of  little  or  no  use,  except  in  finding  a  meridian  line,  or 
the  proper  situation  of  the  land. 

Over  the  brass  circle  is  a  pair  of  sights,  fixed  to  a  moveable 
index,  which  turns  on  the  centre  of  the  instrument,  and  upon 
which  the  circumferentor-box  is  placed. 

'This  instrument  will  either  give  the  angles  of  the  field  or  the 
bearing  of  every  stationary  distance  line  from  the  meridian,  as 
the  circumferentor  and  quartered  compass  do. 

To  take  the  angles  of  the  field. 
PL.  6.  Jig.  6. 

Lay  the  ends  of  your  index  io  360°  and  180°  ;  ^turn  the  whole 
about  with  the  360  from  you ;  direct  the  sights  from  A  to  G9 
and  screw  the  instrument  fast ;  direct  them  from  A  to  cut  the 
object  at  B ;  the  degree  then  cut  by  that  end  of  the  index  which 
is  opposite  you  will  be  the  quantity  of  the  angle  GAB  to  place 
in  your  field-book  ;  to  which  annex  the  measure  of  the  line  AB 
in  chains  and  links ;  set  up  your  instrument  at  JB,  unscrew  it, 
and  lay  the  ends  of  your  index  to  360  and  180  ;  turn  the  whole 
about,  with  the  360  from  you  or  180  next  you,  till  you  cut  the 
object  at  A ;  screw  the  instrument  fast  and  direct  your  sights  to 
the  object  at  C,  and  the  degree  then  cut  by  that  end  of  the  index 
which  is  opposite  to  you  will  be  the  quantity  of  the  angle  ABC. 


126  THE  THEODOLITE. 

Thus  proceed  from  station  to  station,  still  laying  the  index  to 
360,  turning  it  from  you,  and  observing  the  object  at  the  fore- 
going station,  screwing  the  instrument  fast  and  observing  the 
object  at  the  following  station,  and  counting  the  degrees  to 
the  opposite  end  of  the  index,  will  give  you  the  quantity  of  each 
respective  angle. 

LEMMA. 

All  the  angles  of  any  polygon  are  equal  to  iwice  as  many  right  angles  as 
there  are  sides,  less  by  four.  Thus,  all  the  angles  A,  B,  C,  D,  E,  F,  £, 
are  equal  to  twice  as  many  right  angles  as  there  are  sides  in  the  figure,  les* 
by  four. 

PL.  G.Jig.  6. 

Let  the  polygon  be  disposed  into  triangles  by  lines  drawn 
from  any  assigned  point  //  within  it,  as  by  the  lines  HA,  HB, 
HC,  &LC.  It  is  evident,  then  (by  theo.  2,  sect.  4,  part  1),  that 
the  three  angle's  of  each  triangle  are  equal  to  two  right,  and 
consequently  that  the  angles  in  all  the  triangles  are  twice  as 
many  right  ones  as  there  are  sides  :  but  all  the  angles  about 
the  point  H  are  equal  to  four  right  (by  cor.  2,  theo.  1,  sect. 
4) ;  therefore  the  remaining  angles  are  equal  to  twice  as  many 
right  ones  as  there  are  sides  in  the  figure,  abating  four. 
Q.  E.  D. 

SCHOLIUM. 

Hence  we  may  know  if  the  angles  of  a  survey  be  truly  taken  ; 
for  if  their  sum  be  equal  to  twice  as  many  right  angles  as  there 
are  stations,  abating  four  right  angles,  you  may  conclude  that 
the  angles  were  truly  taken,  otherwise  not. 

If  you  take  the  bearing  of  any  line  with  the  circumferentor, 
that  bearing  will  be  the  number  of  degrees  the  line  is  from  the 
north ;  consequently  the  north  must  be  a  like  number  of  de- 
grees from  the  line  ;  and  thus  the  north,  and  of  course  the  south, 
as  well  as  the  east  and  west,  or  the  situation  of  the  land,  is  ob- 
tained. \ 

To  take  the  bearing  of  each  respective  line  from  the  meridian ;  or  to 
perform  the  office  of  the  circumferentor,  or  quartered,  compass,  by  the  the" 
odolite. 

Set  your  instrument  at  the  first  station,  and  lay  the  index  to 
360°  and  180°  with  the  flower-de-luce  of  the  box  next  360; 
unscrew  the  instrument,  and  turn  the  whole  about  till  the 
north  and  south  points  of  the  needle  cut  the  north  and  south 
points  in  the  box ;  then  screw  it  fast,  and  the  instrument  is 
north  and  south,  if  there  be  no  variation  in  the  needle ;  but  if 
there,  be,  and  its  quantity  known,  it  may  be  easily  allowed. 


THE  THEODOLITE.  127 

The  circuraferentor-box  may  then  be  taken  off. 

Direct  the  sights  to  the  object  at  the  second  station,  and  the 
degree  cut  by  the  opposite  end  of  the  index  will  be  the  bearing 
of  that  line  from  the  north,  and  the  same  that  the  circumferentor 
would  give. 

After  having  measured  the  stationary  distance,  set  up  your 
instrument  at  the  second  station ;  unscrew  it,  and  set  either 
end  of  the  index  to  the  degree  of  the  last  line,  and  turning  the 
whole  about  with  that  degree  towards  you,  direct  your  sights 
to  an  object  at  the  foregoing  station,  and  screw  the  instru- 
ment fast ;  it  will  then  be  parallel  to  its  former  situation,  and 
consequently  north  and  south  ;  direct  then  your  sights  to  an 
object  at  the  following  station,  and  the  degree  cut  by  the  oppo- 
site end  of  the  index  will  be  the  bearing  of  that  line. 

In  the"  like  manner  you  may  proceed  through  the  whole. 

If  the  brass  circle  be  divided  into  four  nineties,  from  360  and 
180,  and  the  letters  N,  S,  £,  VFbe  applied  to  them,  the  bear- 
ings may  be  obtained  by  putting  down  the  letters  the  far  or  op- 
posite end  of  the  index  lies  between,  and  annexing  thereto  the 
degrees  from  the  N  or  S,  and  this  is  the  same  as  the  quartered 
compass. 

If  you  keep  the  compass-box  on,  to  see  the  mutual  agreenlent 
of  the  two  instruments :  after  having  fixed  the  theodolite  north 
and  south,  as  before,  turn  the  index  about,  the  north  end  or  flower- 
de-luce  next  your  eye,  and  count  the  degree  to  the  opposite  or 
south  end  of  the  index,  and  this  will  correspond  with  the  de- 
gree cut  by  the  south  end  of  the  needle. 

At  the  second  or  next  'station,  unscrew  the  instrument  and 
set  the  south  of  the  index  to  the  degree  of  the  last  station ;  turn 
the  whole  about,  with  the  south  of  the  index  to  you,  and  cut  the 
object  at  the  foregoing  station ;  screw  the  instrument  fast,  and 
with  the  north  of  the  index  to  you,  cut  the  object  at  the  next 
following  station  ;  the  degree  then  cut  by  the  south  of  the  index 
will  correspond  with  the  degree  cut  by  the  south  end  of  the 
needle,  and  so  through  the  whole. 

Some  theodolites  have  a  standing  pair  of  sights  fixed  at  360 
and  180,  besides  those  on  the  moveable  index;  if  you  would 
use  both,  look  through  the  standing  sights  with  the  180  next 
you  to  an  object  at  the  foregoing  station  :  screw  the  instrument 
fast,  and  direct  the  upper  sights  on  the  moveable  index  to  the 
object  at  the  following  station,  and  the  degree  cut  by  the  oppo- 
site end  of  the  index  will  give  you  the  quantity  of  the  angle  of 
the  field. 

Two  pair  of  sights  can  be  of  no  use  in  finding  the  angles 
from  the  meridian ;  and  inasmuch  as  one  pair  is  sufficient  to 


128         THE  SEMICIRCLE— PLANE  TABLE. 

find  the  angles  of  the  field,  the  second  can  be  of  no  use :  be- 
sides, they  obstruct  the  free  motion  of  the  moveable  index,  and 
therefore  are  rather  an  incumbrance  than  of  any  real  use. 

Some  will  have  it  that  they  are  useful  with  the  others  for 
setting  off  a  right  angle  in  taking  an  offset :  and  surely  this  is 
as  easily  performed  by  the  one  pair  on  the  moveable  index  : 
thus,  if  you  lay  the  index  to  360  and  180,  and  cut  the  object 
either  in  the  last  or  following  station,  screw  the  instrument  fast 
and  turn  the  index  to  90  and  270,  and  then  it  will  be  at  right 
angles  with  the  line.  So  that  the  small  sights,  at  those  of  the 
circle,  can  be  of  no  additional  use  to  the  instrument,  and  there- 
fore should  be  laid  aside  as  useless. 

This  instrument  may  be  used  in  windy  and  rainy  weather, 
as  well  as  in  mountainous  and  hilly  grounds;  for  it  does  not 
require  a  horizontal  position  to  find  the  bearing  or  angle,  as  the 
needle  doth,, and  therefore  is  preferred  to  any  instrument  that  is 
governed  by  the  needle. 


THE  SEMICIRCLE. 

THIS  instrument,  as  its  name  imports,  is  a  half-circle,  divided 
from  its  diameter  into  180  degrees,  and  from  thence  again,  that 
is,  from  0  to  360  degrees.  It  is  generally  made  of  brass,  and  is 
from  8  to  18  inches  diameter. 

On  the  centre  there  is  a  moveable  index  with  sights,  on 
which  is  placed  a  circumferentor-box,  as  in  the  theodolite. 

This  instrument  may  be  used  as  the  theodolite  in  all  re- 
spects, but  with  this  difference ;  when  you  are  to  reckon  the 
degree  to  that  end  of  the  index  which  is  off  the  semicircle,  you 
may  find  it  at  the  other  end,  reckoning  the  degree  from  180 
forwards. 


THE  PLANE  TABLE.* 

A  PLANE  TABLE  is  an  oblong  of  oak,  or  other  wood,  about 
15  inches  long  and  12  broad.  They  are  generally  composed  of 
three  boards,  which  are  easily  taken  asunder  or  put  together 
for  the  convenience  of  carriage. 

*  This  instrument  is  not  much  used  by  surveyors  at  present. 


THE  PLANE  TABLE.  120 

There  is  a  box  frame,  with  six  joints  in  it,  to  take  off  and  put 
on  as  occasion  serves ;  it  keeps  the  table  together,  and  is  like- 
wise of  use  to  keep  down  a  sheet  of  paper  which  is  put  thereon. 

The  outside  of  the  frame  is  divided  into  inches  and  tenths, 
which  serve  for  ruling  parallels  or  squares  on  the  paper,  or  for 
shifting  it,  when  occasion  serves. 

The  inside  of  the  frame  is  divided  into  360  degrees,  which, 
though  unequal  on  it,  yet  are  the  degrees  of  a  circle  produced 
from  its  centre,  or  centre  of  the  table,  where  there  is  a  small 
hole. 

The  degrees  are  subdivided  as  small  as  their  distance  will 
admit ;  at  every  tenth  degree  are  two  numbers,  one  the  number 
of  degrees,  the  other  its  complement,  to  360. 

There  is  another  centre-hole  about  one-fourth  of  the  table's 
breadth  from  one  edge,  and  is  in  the  middle  between  the  two  ends. 
To  this  centre-hole  on  the  other  side  of  the  frame,  there  are  the 
divisions  of  a  semicircle,  or  180  degrees ;  and  these  again  are 
subdivided  into  halves,  or  quarters,  as  the  size  of  the  instru- 
ment will  admit. 

That  side  of  the  frame  on  which  the  360  degrees  are,  sup- 
plies the  place  of  a  theodolite,  the  other  that  of  a  semicircle. 

There  is  a  circumferentor-box  of  wood,  with  a  paper  chart 
at  the  bottom,  applied  to  one  side  of  the  table  by  a  dovetail 
joint  fastened  by  a  screw.  This  box  (besides  its  rendering  the 
plane  table  capable  of  answering  the  end  of  a  circumferentor) 
is  very  useful  for  placing  the  instrument  in  the  same  position 
every  remove. 

There  is  a  brass  ruler  or  index,  about  two  inches  broad, 
with  a  sharp  of  fiducial  edge,  at  each  end  of  which  is  a  sight ; 
on  the  ruler  are  scales  of  equal  parts,  with  and  without  diago- 
nals, and  a  scale  of  chords ;  the  whole  is  fixed  on  a  ball  and 
socket,  and  set  on  a  three-legged  staff. 

To  take  the  angles  of  afield  by  the  table. 

Having  placed  the  instrument  at  the  first  station,  turn  it  about 
till  the  north  end  of  the  needle  be  over  the  meridian,  or  flower- 
de-luce  of  the  box,  and  there  screw  it  fast.  Assign  any  con- 
venient point,  to  which  apply  the  edge  of  the  index,  so  as 
through  the  sights  you  may  see  the  object  in  the  last  station, 
and  by  the  edge  of  the  index  from  the  point  draw  a  line.  Again, 
turn  about  the  index  with  its  edge  to  the  same  point,  and  through 
the  sights  observe  the  object  in  the  second  station,  and  from 
the  point,  by  the  edge  of  the  index,  draw  another  line ;  so  is 
the  angle  laid  down ;  on  that  last  line  set  off  the  distance  to 
the  second  station,  in  chains  and  links ;  apply  your  instrument 

F3 


130  THE  PLANE  TABLE. 

to  the  second  station,  taking  the  angle  as  before  ;  and  after  the 
like  manner  proceed  till  the  whole  is  finished. 

This  method  may  be  used  in  good  weather,  if  the  needle  be 
well  touched  and  play  freely;  but  if  it  be  in  windy  weather,  or 
the  needle  out  of  order,  it  is  better,  after  having  taken  the  first 
angle  as  before,  and  having  removed  your  instrument  to  the 
second  station,  and  placed  the  needle  over  the  meridian  line  as 
before,  to  lay  the  index  on  the  last  drawn  line,  and  look  back- 
ward through  the  sights  ;  if  you  then  see  the  object  in  the  first 
station,  the  tafcle  is  fixed  right,  and  the  needle  is  true ;  if  not, 
turn  the  table  about,  the  index  lying  on  the  last  line,  till  through 
the  sights  you  see  the  object  in  the  first  station  :  and  then  screw 
it  fast,  and  keeping  the  edge  of  the  index  to  the  second  station, 
direct  your  sights  to  the  next ;  draw  a  line  by  the  edge  of  the 
index,  and  lay  off  the  next  line ;  and  proceed  through  the 
whole  without  using  the  needle,  as  you  do  with  the  theodolite. 

If  the  sheet  of  paper  on  the  table  be  not  large  enough  to  con- 
tain the  map  of  the  ground  you  survey,  you  must  put  on  a  clean 
sheet,  when  the  other  is  full ;  and  this  is  called  shifting  of 
paper,  and  is  thus  performed.. 

PL.  6.^.8. 

Let  ABCD  represent  the  sheet  of  paper  on  the  plane  table, 
upon  which  the  plot  £,  jP,  Gf,  JFT,  /,  K,  L,  M  is  to  be  drawn : 
let  the  first  station  be  E ;  proceed  as  before,  from  thence  to  F 
and  to  G ;  then  proceeding  to  I/,  you  find  there  is  not  room  on 
your  paper  for  the  line  GH,  however  draw  as  much  of  the  line 
GH  as  the  paper  can  hold,  or  draw  it  to  the  paper's  edge. 
Move  your  instrument  back  to  the  first  station  JS,  and  pro- 
ceed the  contrary  way  to  M  and  to  L  ;  but  in  going  from  thence 
to  K,  you  again  find  your  sheet  will  not  hold  it ;  however  draw 
as  much  of  the  line  LK  on  the  sheet  as  it  can  hold. 
!  Take  that  sheet  off  the  table,  first  observing  the  distance  oo 
of  the  lines  GH  an<!  LK  by  the  edge  of  the  table  ;  take  off 
that  sheet  and  mark  it  with  No.  1,  to  signify  it  to  be  the  first 
taken  off.  Having  then  put  on  another  sheet,  lay  that  distance 

00  on  the  contrary  end  of  the  table,  and  so  proceed  as  before 
with  the  residue  of  the  survey,  from  o  to  H,  to  K,  and  thence 
to  o ;  so  is  your  survey  complete. 

1  In  the  like  manner  you  may  proceed  to  take  off  and  put  on 
as  many  sheets  as  are  convenient ;  and  these  may  afterward 
be  joined  together  with  mouth  glue,  or  fine  white  wafer,  very 
thin. 

If  the  index  be  fixed  to  the  first  centre,  using  the  360  side,  it 
;will  then  serve  as  a  theodolite,  and  when  to  the  second  centre, 


OF  ANGLES  OF  ELEVATION,  &c.  131 

using  the  180  side,  it  will  serve  as  a  semicircle  ;  by  either  of 
which  you  may  survey  in  rainy  weather,  when  you  cannot  have 
paper  on  the  lable. 


To  measure  Angles  of  Altitude  by  the  Circumferentor,  Theodo- 
lite, Semicircle,  or  Plane  Table. 

1.  To  take  an  angle  of  altitude  by  the  circumferentor,  let 
the  glass  lid  be  taken  off,  and  let  the  instrument  be  turned  on 
one  side,  with  the  stem  of  the  ball  into  the  notch  of  the  socket, 
so  that  the  circle  may  be  perpendicular  to  the  plane  of  the 
horizon ;  let  the  instrument  be  placed  in  this  situation  before 
the  object,  so  that  the  top  thereof  may  be  seen  through  the 
sights ;  let  a  plummet  be  suspended  from  the  centre-pin,  and 
the  object  being  then  'observed,  the  complement  of  the  number 
of  degrees  comprehended  between  the  thread  of  the  plummet 
and  that  part  of  the  instrument  which  is  next  your  eye  will  give 
the  angle  of  altitude  required. 

2.  If  an  angle  of  altitude  is  to  be  taken  by  the  theodolite, 
or  semicircle,  let  a  thread  be  'run  through  a  hole  at  the  centre, 
and  a  plummet  be  suspended  by  it ;  turn  the  instrument  on  one 
side,  by  the  help  of  the  ball  and  notch  in  the  socket  for  that 
purpose,  so  that  the  thread  may  cut  90,  having  360  degrees  next 
you ;  screw  it  fast  in  that  position,  and  through  the  sights  cut 
the  top  of  the  objects  ;  and  the  degrees  then  cut  by  the  end  of 
the  index  next  you-  are  the  degrees  of  elevation  required.     An 
angle  of  depression  is  taken  the  contrary  way. 

3.  By  the  plane  table  an  angle  of  altitude  is  taken  in  the 
like  manner ;  by  suspending  a  plummet  from  the  centre  thereof 
having  turned  the  table  on  one  side,  and  fixed  the  index  to  the 
centre  by  a  screw,  so  as  to  move  freely,  let  the  thread  cut  90, 
look  through  the  sights  as  before,  and  you  have  the  angle  of 
elevation,  and  on  the  contrary  that  of  depression. 


THE  PROTRACTOR. 

THE  protractor  is  a  semicircle  annexed  to  a  scale,  and  is 
iriade  of  brass,  ivory,  or  horn  \  its  diameter  is  generally  about 
five  or  six  inches. 

The  semicircle  contains  three  concentric  semicircles,  at  such 
distances  from  each  other  that  the  spaces  between  them  may 
contain  figures. 


132 


THE  PROTRACTOR. 


The  outward  circle  is  numbered  from  the  right  to  the  left- 
hand,  with  10,  20,  30,  &c.  to  180  degrees  ;  the  middlemost  the 
same  way,  from  180  to  360  degrees  ;  and  the  innermost  from 
the  upper  edge  of  the  scale  both  ways,  from  10,  20,  30,  &c.  to 
90  degrees. 

It  is  easy  to  conceive  that  the  protractor,  though  a  semicircle, 
may  be  made  to  supply  the  place  of  a  whole  circle  ;  for  if  a 
line  be  drawn,  and  the  centre-hole  of  the  protractor  be  laid  on 
any  point  in  that  line,  the  upper  edge  of  the  scale  corresponding 
with  that  line,  the  divisions  on  the  edge  of  the  semicircle  will 
run  from  0  to  180,  from  right  to  left  :  again,  if  it  be  turned  the 
other  way,  or  downwards,  keeping  the  centre-hole  thereof  on 
the  aforesaid  point  in  the  line,  then  the  divisions-  will  run  from 
180  to  360,  and  so  complete  an  entire  circle  with  the  former 
semicircle. 

The  use  of  the  protractor  is  to  lay  ofi^  angles,  and  to  de- 
lineate or  draw  a  map  or  plan  of  any  ground  from  the  field- 
notes  ;  and  is  performed  in  the  following  manner. 

'  To  protract  afield-hfok,  when  the  angles  are  taken  from  the  meridian. 

.  9. 


On  your  paper  rule  lines  parallel  to  each  other,  at  an  inch 
asunder  (being  most  usual),  or  at  any  other  convenient  dis- 
tance ;  on  the  left  end  of  the  parallels  put  N  for  north,  and  on 
the  right  S  for  south  ;  put  E  at  the  top  for  east,  and  W  at  the 
bottom  of  your  paper  for  west. 

Then  let  the  following  field-book  be  that  which  is  to  be  pro- 
tracted, the  bearings  being  taken  from  the  meridian,  whether 
by  a  circumferentor,  theodolite,  or  semicircle,  and  measured 
with  a  two-pole  chain. 


No. 

Bearing. 

eh.  L 

1 
2 
3 
4 
5 
6 
7 

317* 
266 
193 
124 
63f 

55.  20 
12.  36 
29.  20 
55.  20 
40.  00 
76.  00 
87.  02 

Close  at  the  first  station. 


Pitch  upon  any  convenient  point  on  your  paper  for  your 
first  station,  as  at  1,  on  which  lay  the  centre-hole  of  your  pro- 


THE   PROTRACTOR.  133 

tractor  with  a  protrac ting-pin  ;  then,  if  the  degrees  be  less  than 
180,  turn  the  arc  of  your  protractor  downwards,  or  towards  the 
west,  but  if  more  than  180  upwards  or  towards  the  east. 

Or,  if  the  right-hand  be  made  the.north  and  the  left  the  south, 
the  west  will  be  then  up  and  the  east  down. 

In  this  case,  if  the  degree  be  less  than  180,  turn  the  arc  of 
your  protractor  upwards,  or  towards  the  west ;  and  if  more, 
downwards,  or  towards  the  east. 

By  the  foregoing  field-book  the  first  bearing  is  283| ;  turn 
the  arc  of  your  protractor  upwards,  keeping  the  pin  hi  the  centre- 
hole,  move  the  protractor  so  that  the  parallel  lines  may  cut 
opposite  divisions  either  on  the  ends  of  the  scale  or  on  the  de- 
grees, and  then  it  is  parallel.  This  must  be  always  first  done, 
before  you  lay  off  your  degrees. 

Then  by  the  edge  of  the  semicircle,  keeping  the  protractor 
steady,  with  the  pin  prick  the  first  bearing  283£,  and  from  the 
centre-point,  through  that  point  or  prick,  draw  a  blank  line  with 
the  pin,  on  which,  from  a  scale  of  equal  parts  or  from  the  scale's 
edge  of  the  protractor,  lay  off  the  distance  55ch.  201. ;  so  is 
that  station  protracted. 

At  the  end  of  the  first  station,  or  at  2,  which  is  the  beginning 
of  the  second,  with  the  pin  place  the  centre  of  the  protractor, 
turning  the  arc  up,  because  the  bearing  of  the  second  station  is 
more  than  180,  viz.  348f.  Place  your  protractor  parallel,  as 
before,  and,  by  the  edge  of  the  semicircle,  with  the  pin  prick 
at  that  degree,  through  which  and  the  end  of  the  foregoing 
station  draw  a  blank  line,  and  on  it  set  the  distance  of  that 
station.  \ 

In  the  like  manner  proceed  through  the  whole,  only  observe 
to  turn  the  arc  of  your  protractor  down  when  the  degrees  are 
less  than  180. 

If  you  lay  off  the  stationary  distances  by  the  edge  of  the 
protractor,  it  is  necessary  to  observe,  that  if  your  map  is  to  be 
laid  down  by  a  scale  of  40  perches  to  an  inch,  every  division 
on  the  protractor's  edge  will  be  one  two-pole  chain  ;  1  of  a  di- 
vision will  be  25  links,  and  {  of  a  division  will  be  1 2£  links. 

If  your  map  is  to  be  laid  down  by  a  scale  of  20  perches  to 
an  inch,  two  divisions  will  be  one  two-pole  chain ;  one  division 
will  be  25  links ;  £  a  division  12£  links  ;  and  £  of  a  division  will 
be  6i  links. 

In  general,  if  25  links  be  multiplied  by  the  number  of  perches 
to  an  inch  the  map  is  to  be  laid  down  by,  and  the  product  be 
divided  by  20  (or,  which  is  the  same  thing,  if  you  cut  off  one  and 
take  the  half),  you  will  have  the  value  of  one  division  on  the 
protractor's  edge  hi  links  and  parts.  .  . 


'!&  THE  PROTRACTOR. 

t:.-i.  EXAMPLES. 

1.    How  many  links  in  a  division,  if  a  map  be  J&id  down  by 
a  scale  of  8  perches  to  an  inch  ? 

25 
8 

210)20(0 

10  links,  Answer. 

1     2.  How  many  links  in  a  division,  if  a  map  be  laid  down  by 
a  scale  of  10  perches  to  an  inch  I 
25 
10 

,210)25)0 

12.5  or  121  links,  Answer. 
And  so  of  any  other.        • 

To  protract  afield-book  taken  ly  the  angles  of  the  field. 

Note. — We  here  suppose  .the  land  surveyed  is  kept  on  the 
right-hand  as  you  survey. 

Draw  a  blank  line  with  a  ruler  of  a  length  greater  than  the 
diameter  of  the  protractor ;  pitch  upon  any  convenient  point 
therein,  to  which  apply  the  centre-hole  of  your  protractor  with 
your  pin,  turning  the  arc  upwards  if  the  angle  be  less  than  180, 
and  downwards  if  more ;  and  observe  to  keep  the  upper  edge 
of  the  scale,  or  1 80  and  0  degrees,  upon  the  line :  then  prick 
off  the  number  of  degrees  contained  in  the  given  angle,  and  draw 
a  line  from  the  first  point  through  the  point  at  the  degrees, 
upon  which  lay  the  stationary  distance.  Let  this  line  be  length- 
ened forwards  and  backwards,  keeping  your  first  station  to  the 
right  and  second  to  the  left,  and  lay  the  centre  of  your  pro- 
tractor over  the  second  station,  with  your  pin  turning  the  arc 
upwards  if  the  angle  be  less  than  180,  and  downwards  if  more ; 
and  keeping  the  180  and  0  degrees  on  the  line,  prick  off  the 
number  of  degrees  contained  in  the  given  angle,  and  through 
that  point  and  the  last  station  draw  a  line,  on  which  lay  the 
stationary  distance ;  and  in  like  manner  proceed  through  the 
whole. 

In  all  protractions,  if  the  end  of  the  last  station  falls  exactly 
in  the  point  you  began  at, -the  field-work  and  protraction  are  truly 


LIST  OF  INSTRUMENTS.  13S 

taken  and  performed ;  if  not,  an  error  must  have  been  commit- 
ted in  one  of  them :  in  such  case,  make  a  second  protraction ; 
if  this  agrees  with  the  former,  and  neither  meet  nor  close,  the 
fault  is  in  the  field-work  and  not  in  the  protraction  and  then 
a  resurvey  must  be  taken. 

REMARKS. 

The  accuracy  of  geometrical  and  trigonometrical  mensuration 
depends  in  a  great  degree  on  the  exactness  and  perfection  of 
the  instruments  made  use  of;  if  these  are  defective  in  construc- 
tion or  difficult  in  use  the  surveyor  will  either  be  subject  to  er- 
ror or  embarrassed  with  continual  obstacles.  If  the  adjustments 
by  which  they  are  to  be  rendered  fit  for  observation  be  trouble- 
some and  inconvenient,  they  will  be  taken  upon  trust,  and  the 
instrument  will  be  used  without  examination,  and  thus  subject 
the  surveyor  to  errors  that  he  can  neither  account  for  nor 
correct. 

In  the  present  state  of  science  it  may  be  laid  down  as  a 
maxim,  that  every  instrument  should  be  so  contrived  that  the 
observer  may  easily  examine  and  rectify  the  principal  parts ; 
for  however  careful  the  instrument-maker  may  be,  however  per- 
fect the  execution  thereof,  it  is  not  possible  that  any  instrument 
should  long  remain  accurately  fixed  in  the  position  in  which 
it  came  out  of  the  maker's  hand,  and  therefore  the  principal 
parts  should  be  moveable,  to  be  rectified  occasionally  by  the 

observer.  \ 

• 

An  enumeration  of  Instruments  useful  to  a  Surveyor, 

Fewer  or  more  of  which  will  be  wanted,  according  to  the  ex- 
tent of  his  work  and  the  accuracy  required. 
A  case  of  good  pocket  instruments. 
A  pair  of  beam  compasses. 
A  set  of  feather-edged  plotting  scales. 
Three  or  four  parallel  rules. 
A  pair  of  proportional  compasses. 
A  pair  of  triangular  ditto. 
A  pentagraph. 

A  cross-staff.  • 

A  circumferentor. 
A  Hadley's  sextant. 
An  artificial  horizon.  - 

A  theodolite. 
A  surveying  compass. 
Measuring  chains  and  measuring  tapes. 


r 


136  LIST  OF  INSTRUMENTS. 

King's  surveying  quadrant 
A  perambulator,  or  measuring  wheel. 
A  spirit-level  with  telescope. 
Station  staves  used  with  the  level. 
A  protractor,  with  or  without  a  nonius. 

To  be  added  for  County  and  Marine  Surveying. 

An  astronomical  quadrant,  or  circular  instrument. 
A  good  refracting  and  reflecting  telescope. 
A  copying-glass. 

For  Marine  Surveying. 

A  station-pointer. 

An  azimuth  compass. 

One  or  two  boat  compasses. 

Besides  these,  a  number  of  measuring  rods,  iron  pins,  or  ar- 
rows, &c.  will  be  found  very  convenient,  and  two  or  three  off- 
set staves,  which  are  straight  pieces  of  wood  six  feet  seven  inches 
long,  and  about  an  inch  and  a  quarter  square  :  they  should  be  ac- 
curately divided  into  ten  equal  parts,  each  of  which  will  be  equal 
to  one  links  These  are  used  for  measuring  offsets  and  to  ex- 
amine and  adjust  the  chain. 

Five  or  six  staves,  of  about  five  feet  in  length  and  one  inch 
and  a  half  in  diameter,  the  upper  part  painted  white,  the  lo\ver 
end  shod  with  iron,  to  be  struck  into  the  ground,  as  marks. 

Twenty  or  more  iron  arrows,  ten  of  which  are  always 
wanted  to  use  with  flie  chain,  to  count  the  number  of  links,  and 
preserve  the  direction  of  the  chain,  so  that  the  distance  measured 
may  be  really  in  a  straight  line. 

The  pocket  measuring  tapes,  in  leather  boxes,  are  often  very 
convenient  and  useful.  They  are  made  to  the  different  lengths 
of  one,  two,  three,  four  poles,  or  sixty-six  feet  and  one  hundred 
feet :  divided  on  one  side  into  feet  and  inches,  and  on  the  other  into 
links  of  the  chain.  Instead  of  the  latter  are  sometimes  placed 
the  centesimals  of  a  yard,  or  three  feet  into  100  equal  parts. 


OF  HEIGHTS.  137 

SECTION  II. 
MENSURATION  OF  HEIGHTS  AND  DISTANCES. 

1st.     Of  Heights. 
PL.  5.  fig.  IS. 

THE  instrument  of  least  expense  for  taking  heights,  is  a 
quadrant  divided  into  ninety  equal  parts  or  degrees ;  and  those 
may  be  subdivided  into  halves,  quarters,  or  eighths,  according  to 
the  radius,  or  size  of  the  instrument :  its  construction  will  be 
evident  by  the  scheme  thereof. 

From  the  centre  of  the  quadrant  let  a  plummet  be  suspended 
by  a  horse-hair,  or  a  fine  silk  thread,  of  such  a  length  that  it 
may  vibrate  freely,  near  the  edge  of  its  arc  ;  by  looking  along 
the  edge  AC,  to  the  top  of  the  object  whose  height  is  required, 
and  holding  it  perpendicular,  so  that  the  plummet  may  neither 
swing  from  it,  nor  lie  on  it,  the  degree  then  cut  by  the  hair  or 
thread  will  be  the  angle  or  altitude  required. 

If  the  quadrant  be  fixed  upon  a  ball  and  socket  on  the  three- 
legged  staff,  and  if  the  stem  from  the  ball  be  turned  into  the 
notch  of  the  socket,  so  as  to  bring  the  instrument  into  a  per- 
pendicular position,  the  angle  of  altitude  by  this  means  can  be 
acquired  with  much  greater  certainty. 

An  angle  of  altitude  may  be  also  taken  by  any  of  the  instru- 
ments used  in  surveying ;  as  has  been  particularly  shown  in 
treating  of  their  description  and  uses. 

Most  quadrants  have  a  pair  of  sights  fixed  on  the  edge  A  C9 
with  small  circular  holes  in  them  ;  which  are  useful  in  taking 
the  sun's  altitude,  requisite  to  be  known  in  many  astronomical 
cases ;  this  is  effected  by  letting  the  sun's  ray,  which  passes 
through  the  upper  sight,  fall  upon  the  hole  in  the  lower  one  ; 
and  the  degree  then  cut  by  the  thread  will  be  the  angle  of  the 
sun's  altitude  ;  but  those  sights  are  useless  for  our  present  pur- 
pose,— for  looking  along  the  quadrant's  edge  to  the  top  of  the 
object  will  be  sufficient  as  before. 

To  take  an  angle  of  altitude  and  depression  vrith  the  quadrant. 
PL.  14.^.6,  7. 

Let  A  be  any  object,  as  the  sun,  moon,  or  a  star,  or  the  top 
of  a  tower,  hill,  or  other  eminence  :  and  let  it  be  required  to 
find  the  angle  ABC,  which  aline  drawn  from  the  object  makes 
above  the  horizontil  line  BC. 

Place  the  centre  of  the  quadrant  in  the  angular  point,  and 


138  OF  HEIGHTS. 

move  it  round  there  as  a  centre,  till  with  one  eye  at  D,  the 
other  being  shut,  you  perceive  the  object  A  through  the  sights ; 
then  will  the  arc  GH  of  the  quadrant,  cut  off  by  the  plumb- 
line  BH,  be  the  measure  of  the  angle  ABC,  as  required. 

The  angle  ABC  of  depression  of  any  object  A,  below  the 
horizontal  line  BC,  is  taken  in  the  same  manner  ;  except  that 
here  the  eye  is  applied  to  the  centre,  and  the  measure  of  the 
angle  is  the  arc  GH,  on  the  other  side  of  the  plumb-line.* 

Demonstration.  In  taking  the  angle  of  altitude,  the  angle 
ABG  is  a  right  angle  ;  and  because  the  plumb-line  BH  is  per- 
pendicular to  the  horizon,  the  angle  CBHis  also  a  right  angle ; 
hence  if  the  angle  CBG  be  taken  from  these'  equals,  the  re- 
maining angles  will  be  equal,  that  isABC=GBH,  or  equal  to 
the  arc  HG.  Q.  E.  D. 

In  like  manner,  the  angle  G BH  (in  taking  the  angle  of  de- 
pression) is  equal  to  the  angle  ABC. 

PROBLEM  I. 
PL.  5.  fig.  19. 

To  find  the  height  of  a  perpendicular  object  at  one  station,  which  is  on  a 
horizontal  plane. 

A  steeple. 

["The  angle  of  altitude,  53  degrees. 

p.          J  Distance  from  the  observer  to  the  foot  of  the  stee- 
1      pie,  or  the  base,  85  feet. 
(_  Height  of  the  instrument,  or  of  the  observer,  5  feet. 
Required  the  height  of  the  steeple. 

The  figure  is  constructed  and  wrought,  in  all  respects,  as 
Case  2  of  right-angled  trigonometry ;  only  there  must  be  a  line 
drawn  parallel  to  and  beneath  AB,  of  5  feet,  for  the  observer's 
height,  to  represent  the  plane  upon  which  the  object  stands  ;  to 
which  the  perpendicular  must  be  continued,  and  that  will  be  the 
height  of  the  object. 

Thus,  AB  is  the  base,  A  the  angle  of  altitude,  BC  the  height 
of  the  steeple  from  the  instrument,  or  from  the  observer's  eye, 

*  In  finding  the  height  of  an  object,  let  the  observed  angle  be  as  near  45° 
as  possible,  for  then  a  small  error  committed  in  taking  it  makes  the  least 
error  in  the  computed  height  of  the  object.  In  taking  the  height  of  a 
perpendicular  object,  if  the  observed  angle  be  45°,  the  height  of  the  ob- 
ject above  the  horizontal  line  is  equal  to  the  base  line,  and  if  the  observed 
angle  should  be  60°,  three  times  the  square  of  the  base  line  is  equal  to  the 
square  of  the  perpendicular  object  above  the  horizontal  line ;  hence  by 
extracting  the  square  root  of  three  times  the  square  of  the  base  or  hori 
zontal  line,  will  give  the  height  of  the  object  above  that  line,  to  which  add 
the  height  of  the  observer's  eye  above  the  horizon,  and  you  have  the  true 
height. 


OF  HEIGHTS.  139 

if  he  were  at  the  foot  of  it  ;  DC  the  height  of  the  steeple  above 
the  horizontal  surface. 

Various  statings  for  BC,  as  in  Case  2  of  right-angled  plane 
trigonometry. 

90° 


37=  C. 

l.S.C-.AB:  :  S.A  :  BC. 
37°     85         53°     112.8. 

2.  R.  :AB::  T.A  :  BC. 
90°      85         53°     112.8. 

3.  T.C:  ABi:  R.  :  BC. 
37°      85        90°  112.8. 

ToBC          112.8 

Add  DB  5.     the  height  of  the  observer. 


Their  sum  is  117.8  or  118  feet,  the  height  of  the  steeple  re- 
quired. 

PROBLEM  II. 


To  find  the  height  of  a  perpendicular  object,  on  a  horizontal  plane,  by  having 
the  length  of  the  shadow  given. 

Provide  a  rod,  or  staff,  whose  length  is  given,  let  that  be  set 
perpendicular,  by  the  help  of  a  quadrant,  thus  ;  apply  the  side 
of  the  quadrant  AC  to  the  rod,  or  staff;  and  when  the  thread 
cuts  90°,  it  Is  then  perpendicular;  the  same  may  be  done  by  a 
carpenter's  or  mason's  plumb. 

Having  thus  set  the  rod  or  staff  perpendicular,  measure  the 
length  of  its  shadow  when  the  sun  shines,  as  well  as  the  length 
of  the  shadow  of  the  object  whose  height  is  required,  and  you 
have  the  proper  requisites  given.  Thus, 

ab,  the  length  of  the  shadow  of  the  staff,  15  feet. 

be,  the  length  of  the  staff,  10  feet. 

AB,  the  length  of  the  shadow  of  the  steeple,  or  object,  135  feet. 

Required  BC,  the  height  of  the  object. 

The  triangles  abc,  ABC  are  similar,  thus  ;  the  angle  b=B, 
being  both  right  ;  the  lines  ac,  A  C  are  parallel,  being  rays,  or 
a  ray  of  the  sun  ;  whence  the  angle  a=  A  (by  part  3,  theo.  3, 
sect.  4),  and  consequently  c=  C.  The  triangles,  being  therefore 
mutually  equiangular,  are  similar  (by  theo.  1  6,  sect.  4),  it  will  be, 

ab  :  be  :  :  AB  :  BC. 

15    10       135      90,  the  steeple's  height  required. 


140  OF  HEIGHTS. 

The  foregoing  method  is  most  to  be  depended    ,     however, 
*his  is  mentioned  for  variety's  sake. 


PROBLEM  III. 
PL.  5.  fig.  21. 

To  take  the  altitude  of  a  perpendicular  object  at  t*jfoot  of  a  hill)  from  the 
hill's  side. 

Turn  the  centre  A  of  the  quadrant  next  your  eye,  and  look 
the  ?ide  -AC,  or  90  sidev  to  the  top  and  bottom  of  the 
object;  and  noting  down  the  angles,  measure  the  distance  from 
the  place  of  observation  *o  the  foot  of  the  object.  Thus, 

f  Angle  to  *he  foot  of  the  object,  55i°  or  55°  15'. 
Given  ?  Angle  to  the  top  of  it,  31j°  or  31°  15' 

(  Distance  to  the  foot  of  it,  250  feet. 
Required  .he  height  of  the  object. 

By  Construction. 

Draw  an  indefinite  blank  line  AD  at  any  point  in  which  A 
makes  the  angles  EAB  of  55°  15',  and  EAC  of  31°  15' ;  lay 
250  from  A  to  B ;  from  B  draw  the  perpendicular  BE  (by  prob. 
7  of  geometry)  prossing  AC  in  C:  so  will  EC  be  the  height 
of  the  object  required. 

In  the  triangle  ABC  there  is  given, 

ABE  the  complement  of  EAB  to  90°,  which  is  34°  45'. 

CAB  the  difference  of  the  given  angles  24°  00'. 

The  side  AB  250.     Required  BC. 

This  is  performed  as  Case  2  of  oblique  angular  trigonometry. 
Thus, 

180— the  sum  of  ABE  34°  45'  and  CAB  24°  00'=  ACS 
121°  15'.  Then, 

S.ACB  :AB:i  S.CAB  :  BC. 

121°  15'   250       24°  00'     119,  the  height  required. 

PROBLEM  IV. 
PL.  5.fig.  22. 

To  take  the  altitude  of  a  perpendicular  object  on  the  top  of  a  hill  at  one 
station,  when  the  top  and  bottom  of  it  can  be  seen  from  the  foot  of  the  hill. 

As  in  prob.  1,  take  an  angle  to  the  top  and  another  to  the  bot- 
tom of  the  object,  and  measure  from  the  place  of  observation  to 
the  foot  of  the  object,  and  you  have  all  the  given  requisites. 
Thus, 


OF  HEIGHTS.  141 

A  tower  on  a  hill. 

C  Angle  to  the  bottom  48°  30'. 
Given  J  Angle  to  the  top  67°  00'. 

(  Distance  to  the  foot  of  the  object  136  feet. 
Required  the  height  of  the  object. 

By  Construction. 

Make  the  angle  DAB=4S°  30',  and  lay  136  feet  from  4 
to  B ;  from  B  let  fall  the  perpendicular  BD,  and  that  will  be 
the  height  of  the  hill.  Produce  BD  upwards  by  a  blank  line. 
Again,  at  A  make  the  angle  DAC=Q7°  00'  by  a  blank  line, 
and  from  C,  where  that  crosses  the  perpendicular  produced, 
draw  the  line  CB,  and  that  will  be  the  height  of  the  object  re- 
quired. 

Let  AC  be  drawn. 

In  the  triangle  ABC  there  is  given, 

The  angle  A  CD  the  complement  of  DAC=23°  00' 

CAB  the  difference  between  the  two  given  angles  =  18° 
30'. 

And  the  side  AB  136.     To  find  BC, 

S.C:  AB:  :  S.CAB  :  BC. 
.      23°     130        18°  30'     llOi 

IF  BD  were  wanted  it  is  easily  obtained  by  the  first  case  of 
right-angled  plane  trigonometry. 

PROBLEM  V. 

PL.  5.  fig.  23. 

To  take  an  inaccessible  perpendicular  altitude  on  a  horizontal  plane. 

This  is  done  at  two  stations.  Thus, 
Let  DC  be  a  tower  which  cannot  be  approached,  by  means 
of  a  moat  or  ditch,  nearer  than  B ;  at  B  take  an  angle  of  alti- 
tude to  C :  measure  any  convenient  distance  backward  to  A, 
which  note  down ;  at  A  take  another  angle  to  C ;  so  have  you 
the  given  requisites.  Thus, 

f  First  angle  55°  00'. 
Given  ?  Stationary  distance  87  feet. 

(  Second  angle  37°  00'. 
The  height  of  the  tower  CD  is  required. 

By  Construction. 

Upon  an  indefinite  blank  line  lay  off  the  stationary  distance 
87  from  A  to  B ;  from  B  set  off  your  first,  and  from  A  your 


142  OP  HEIGHTS. 

second  angle ;  from  C,  the  point  of  intersection  of  the  lines 
which  form  these  angles,  let  fall  the  perpendicular  CD ;  and 
that  will  be  the  height  of  the  object  required. 

The  external  angle  CBD  of  the  triangle  ABC  is  equal  to 
the  two  internal  opposite  ones  A  and  ACB  (by  theo.  4,  sect. 
4) :  wherefore  if  one  of  the  internal  opposite  angles  be  taken 
from  the  external  angle,  the  remainder  will  be  the  other  internal 
opposite  one.  Thus, 

CBD  55°— A  37°~ ACB  18°. 

Therefore,  in  the  triangle  ABC  we  have  the  angles  A  and  ACB 
with  the  side  AB  given,  to  find  BC. 

S.ACB-.AB:  :  S.A  :  BC. 
18°          87         37°    169.4. 

Having  found  JBC,  we  have  in  the  triangle  BCD  the  angle 
CBD  55°;  consequently  BCD  35°,  and  BC  169.4,  to  find 
DC. 

This  is  performed  by  the  first  case  of  right-angled  trigonome- 
try three  several  ways.  Thus : 

1.  R:  BC::  S.CBD  :  DC. 
90°  169.4         55°      138.8, 

the  height  required. 

2.  Sec.  CBD  :  BC  :  :  T.CBD  :  DC. 

55°          169.4         55°       139.8, 
the  height  required.        , 

3.  Sec.  BCD  :  BC  :  :  R  :  CD. 

35°        169.4    90°  138.8, 
the  height  required. 

If  BD,  the  breadth  of  the  moat,  were  required,  it  may  also 
be  found  by  three  different  statings,  as  in  the  first  case  of  right- 
angled  plane  trigonometry. 

PROBLEM  VI. 
PL.  5.  Jig.  24. 

Let  JBC,  a  maypole,  whose  height  is  100  feet,  be  broken  at 
D,  the  upper  part  of  which,  DC,  falls  upon  a  horizontal  plane, 
so  that  its  extremity  C  is  34  feet  from  the  bottom  or  foot  of  the 
pole. 

Required  the  segments  BD  and  DC. 

By  Construction. 

Lay  34  feet  from  A  to  B ;  on  B  erect  the  perpendicular  BC 
of  100  feet,  and  draw  A  C;  bisect  AC  .(by  prob.  4,  geom.) 


OF  HEIGHTS*  143 

with  the  perpendicular  line  EF-,  and  from  D,  where  it  cuts  the 
perpendicular  BC,  draw  AD,  which  will  be  the  upper  segment, 
and  DB  will  be  the  lower. 

By  cor.  to  lemma  preceding  theo.  7,  geom.,  AD=DC ;  and 
(by  the  lemma)  the  angle  C=CAD. 

In  the  triangle  ABC  find  C,  as  in  Case  6  of  right-angled 
trigonometry.  Thus, 

1.  BCiRnAB:  T.C=GAD. 

100    90°       34          18°  47'. 

By  theo.  4,  geom.  The  external  angle  ADB=37°  34',  or 
to  twice  the  angle  C ;  i.  e.  to  C  and  GAD. 

Then  in  the  triangle  ABD  there  is  ADB  37°  34',  therefore 
also  its  complement  DAB  52°  26'  and  AB  34  given,  to  find 
AD  and  BD. 

By  the  second  case  of  right-angled  trigonometry : 
2.  S.ADB  :AB::R:ADor  DC. 
37°  34'      34        90°       55.77. 
BC— DC=BD. 
100—55.77=44.23,  required. 

These  may  be  had  from  other  statings,  as  in  the  second  case 
aforesaid. 

PROBLEM  VII. 

PL.  5.^.  25. 
To  take  the  altitude  of  a  perpendicular  object  on  a  hitt,  from  a  plane  beneath  tf. 

This  is  done  at  two  stations.     Thus, 
Let  the  height  DC  of  a  windmill  on  a  hill  be  required. 
From  any  part  of  the  plane  whence  the  foot  of  the  object  .can 
be  seen,  let  angles  be  taken  to  the  foot  and  top ;  measure 
thence  any  convenient  distance  towards  the  object,  and  at  the 
end  thereof  take  another  angle  to  the  top,  and  you  have  the 
proper  requisites.     Thus, 

First  station.  Angle  to  the  foot  DAB  21°  00'. 
Angle  to  the  top  CAB  35°  00'. 
Stationary  distance  AB  104  feet. 

Second  station.     Angle  to  the  top  48°  30'. 
DC  required. 

By  Construction. 

On  an  indefinite  blank  line  lay  the  stationary  distance  AB 
104  feet ;  from  A  set  off  the  second,  and  from  B  the  third  given 
angle  ;  and  from  the  intersecting  point  C  of  the  line  formed  by 


144  OF  HEIGHTS. 

them  let  fall  the  perpendicular  CE ;  from  A  set  off  the  first 
angle,  and  the  line  formed  by  it  will  determine  the  point  D. 
Thus  have  we  the  height  of  the  hill,  as  well  as  that  of  the 
windmill. 

The  angle  CBE — A—ACB,  as  in  the  last  prob. 
In  the  triangle  ABC  find  AC  thus, 

S.ACB  :  AB  :  :  S.ABC  (or  sup,  of  CBE)  :  AC. 
13°  30'  :  104  ::  131°  30'  :  333.6. 

The  angle  CAE— DAE=  CAD. 
The  angle  ACD=AED+EAD,  by  theo.  4. 
In  the  triangle  CAD  find  CD  thus, 

S.ADC:   AC   :  :  S.CAD  :  DC. 

Ill0    :  333.6  :  :      14°      :  86.46  required. 
CjE,  BE,  or  DE  may  be  found  by  other  various  statings, 
as  set  forth  in  the  first  and  second  cases  of  right-angled  trigo- 
nometry. 

PROBLEM  VIII. 
PL.  5.  Jig.  26. 

To  find  the  length  of  an  object  that  stands  obliguely'on  the  top  of  a  hillt 
from  a  plane  beneath. 

Let  CD  be  a  tree  whose  length  is  required. 
This  is  done  at  two  stations. 

Make  a  station  at  5,  from  whence  take  an  angle  to  the  foot 
and  another  to  the  top  of  the  tree  ;  measure  any  convenient  dis- 
tance backward  to  A,  from  whence  also  let  an  angle  be  taken 
to  the  foot  and  another  to  the  top,  and  you  have  the  requisites 
given.  Thus : 

First  station.        Angle  to  the  foot  EBD=36°  30'. 
Angle  to  the  top  EBC=44°  30'. 
Stationary  distance  AB=W4  feet. 
Second  station.     Angle  to  the  foot  EAD=24°  30'. 
Angle  to  the  top  EAC=32°  00'. 
Let  DC  and  DE  be  required. 

The  geometrical  constructions  of  this  and  the  next  problem 
are  omitted,  as  what  has  been  already  said,  and  the  figures,  are 
looked  upon  as  sufficient  helps. 

EBC—A=ACB,  or  44°  30'— 320=120  30',  as  before. 
In  the  triangle  ABC  find  BC  thus, 

1.  S.ACB  :AB::  S.A  :  BC. 
12°  30'     104        32*    254.7. 
EBD—EAD=ADB,  or  36°  30'— 24°  30'=12°  00'. 


OF  HEIGHTS.  145 

In  the  triangle  ADB  find  DB  thus, 

2.  S.ADB  :  AB :  :  S.DAB  :  DB. 
12°  00'     104       24°  30'   207.4. 
CBE—DBE=CBD,  or  44°  30'— 36°  30'=8°  00'. 
In  the  triangle  CBD  there  is  given  CB  254.7,  DB  207.4, 
and  the  angle  CBD  8°  00',  to  find  DC. 

This  is  performed  as  Case  3  of  oblique-angled  trigonometry. 
Thus, 

3.  BC+BD:BC—BD  :  :  T.  of  $(BDC+BCD)  :  T.  of 

462.1  47.3  86°  00' 

$(BDC— BCD). 
55°  40'. 

86°  OO'-f  55°  40  =141°  4Q'=BDC. 
86°  00'— 55°  40'=  30°  2Q'=BCD. 
4.  S.BCD  :  BD  :  :  S.CBD  :  D*C. 

30°  20'     207.4      8°  00'     57.15,  length  of  the  tree. 
To  find  DE  in  the  triangle  DBE,  say 
R.:BD::   S.DBE  :  DE. 
90°  207.4       36°  30'  123.4,  height  of  the  hill. 

PROBLEM  IX. 

To  find  the  height  of  an  inaccessible  object  CD,  on  a  hill  BC,  from  ground 
that  is  not  horizontal. 

PL.  6.  Jig.  I. 

From  any  two  points,  as  G  and  A,  whose  distance  GA  is 
measured,  and  therefore  given ;  let  the  angles  HGD,  BAD, 
BAG,  and  EAG  be  taken ;  because  GH  is  parallel  to  EA 
(by  part  2,  theo.  3,  geom.),  the  angle  HGA=EAG',  therefore 
EAG+HGD=AGD  :  and  (by  cor.  l,theo.  1,  geom.)  180  — 
the  sum  of  EAG  and  BAD=GAD;  and,  by  cor.  1,  theo.  5, 
geom.,  180  —  the  sum  of  the  angles  AGD  and  GAD=GDA : 
thus  we  have  the  angles  of  the  triangle  A GD  and  the  side  AG 
given;  thence  (by  Case  2  of  obi.  ang.  trig.)  ADmJj^be  easily 
found.  The  angle  DAB—  CAB=DAC,  and  90*°— BAD= 
ADC,  and  180°— the  sum  of  DAC  and  ADC=ACD  :  so  have 
we  the  several  angles  of  the  triangle  .4.  CD  given,  and  the  side 
AD ;  whence  (by  Case  2  of  obi.  trig.)  CD  may  be  easily  found. 
We  may  also  find  AC,  which  with  the  angle  BAC  will  give 
CB  the  height  of  the  hill. 

The  solutions  of  the  several  problems  in  heights  and  dis- 
tances by  Gunter's  scale  are  omitted;    because  every  par 
ticular  stating  has  been  already  shown  by  it,  in  Trigonometry. 
'     G 


146  OP  DISTANCES. 


2d.     Of  Distances. 

THE  principal  instruments  used  in  surveying  will  give  the 
angles  or  bearings  of  lines  ;  which  was  particularly  shown 
when  we  treated  of  them. 

PROBLEM  I. 
PL.  6.  fig.  2. 

Let  A  and  B  be  two  houses  on  one  side  of  a  river,  whose 
distance  asunder  is  293  perches :  there  is  a  tower  at  C  on  the 
other  side  of  the  river,  that  makes  an  angle  at  A  with  the  line 
AB  of  53°  20',  and  another  at  B  with  the  line  BA  of  66° 
20' ;  required  the  distance  of  the  tower  from  each  house,  viz. 
ACtmdBC. 

This  is  performed  as  Case  2  of  oblique-angled  trigonometry, 
thus, 

1.  S.C:    AB:  :  S.A:   BC. 
60°  20'  293     53°  20'  270.5. 

2.  S.C:    AB:  :  S.B  :  AC. 
60°  20'  293     66°  20'  308.8. 

PROBLEM  II. 
PL.  6.  fig.  11. 

Let  B  and  C  be  two  houses  whose  direct  distance  asunder, 
JBC,  is  inaccessible  :  however,  it  is  known  that  a  house  at  A  is 
252  perches  from  B,  and  230  from  C,  arid  that  the  angle  BAG 
is  found  to  be  70°.  What  is  the  distance  BC  between  the  two 
houses  ? 

This  is  performed  as  Case  3  of  oblique-angled  trigonometry, 
thus, 

1.  AB+AC-.AB— AC::T.ofi(C+B):  T.  of  J(C— B). 

482  22  55°  00'        3°  44'. 

55°+3°  44'=58°  44'= C.     55°— 3°  44'=51°  16'=jB. 
2.    S.C:  AB:  :  S.A  :  BC. 
58°  44'  252       70°     277. 

PROBLEM  III. 
PL.  6.  fig.  3. 

Suppose  ABC  a  triangular  piece  of  ground,^  which  by  an  old 
survey  we  find  to  be  thus ;  AB  260,  AC  160,  BC  150  perches, 


c* 


OF  DISTANCES.  147 

the  mearing  lines  AC  and  BC  are  destroyed  or  ploughed  down, 
and  the  line  AS  only  remaining.     What  angles  must  be  set  off 
at  -4  and  B  to  run  new  mearings  by  exactly  where  the  old  s 
ones  were  ? 

This  is  performed  as  in  Case  4  of  oblique-angled 
metry,  thus, 

1.    AB  :  AC-\-BC  :  :  AC—BC  :  AD—DB. 
260         310  10  11.92. 

130+5.96=135.96= AD. 
130— 5.96  =  124.04=/)#. 

2.  AD  :  R  :  :  AC  :  sec.  A. 
136    90°    160     31°  47'. 

3.  BC  :  S.A  n  AC  :  S.B. 
150  31°  47'     160  34°  10'. 

PROBLEM  IV. 
PL.  B.fig.  4. 

Let  D  and  C  be  two  trees  in  a  bog,  to  which  you  can  have 
no  nearer  access  than  at  A  and  B  ;  there  is  given  DAB  100°, 
CAB  36°  30',  CBA  121°,  DBA  49°,  and  the  line  AB  113 
perches.  Required  the  distance  of  the  trees  DC. 

180°  — the  sum  of  DBA  and  DAB=ADB=3l°. 
180°  — the  sum  of  CAB  and  CBA=ACB=22°  30'. 

In  the  triangle  ABD,  find  DB  thus, 

1.  S.ADB  :  AB::  S.DAB  :  DB. 

31°       113          100°       216. 

And  in  the  triangle  ABC,  find  BC  thus, 

2.  S.ACB  :AB::  S.CAB  :  BC. 
22°  30'     113        36°  30'     175.6. 

In  the  triangle  DBC,  you  have  DBC=ABC—ABD=72°, 

likewise  the  sides  BD,  BC  as  before  found,  given,  to  find  DC. 

3.    BD+BC  :  BD—BC  :  :  T.  of  ±(DCB+CDB)  :  T. 

391.6  40.4  54° 

of  $(DCB—CDB). 
8°  05'. 

54° 4-8°  05'=62°  05'=DCB. 
54°— 8°  05'=45°  55'=CDB. 
4.  S.CDB  :  BC::  S.DBC  :  DC. 
45°  55'  175.6         72°     232.5. 
G2 


148  OF  DISTANCES. 

LEMMA. 

»     PL.  6.  fa.  10. 


a  point  C  of  a  triangle  ABC,  inscribed  in  a  circle)  there  be  a  per- 
pendicular CD  let  fall  upon  the  opposite  side  AB,  that  perpendicular  is  to 
one  of  the  sides,  including  the  angle,  as  the  other  side,  including  the  an- 
gle, is  to  the  diameter  of  the  circle,  i.  e.  DC  :  AC  : :  CB  :  CE. 

Let  the  diameter  CE  be  drawn,  and  join  EB ;  it  is  plain,  the 
angle  CEB=CAB(by  cor.  2,  theo.  7,  geom.),  and  CBE  is  a 
right  angle  (by  cor.  5,  theo.  7,  geom.),  and  '=  ADC  :  whence 
ECB=ACD.  The  triangles  CEB,  CAD  are  therefore  mu- 
tually equiangular,  and  (by  theo.  16,  geom'.)  DC  :  AC  :  :  CB  : 
CE,  or  DC:  CB::  AC:  CE.  Q.  E.  D. 

* 

PROBLEM  V. 
PL.  G.fig.  5. 

Let  three  gentlemen's  seats  A,  B,  C  be  situate  in  a  triangular 
form :  there  is  given,  AB  2.5  miles,  AC  2.3,  and  BC  2.  It 
is  required  to  build  a  church  at  E,  that  shall  be  equidistant 
from  the  seats  A,  B,  C.  What  distance  must  it  be  from  each 
seat,  and  by  what  angle  may  the  plaee  of  it  be  found  ? 

By  Construction. 

'By  prob.  15,  geona.,  find  the  centre  of  a  circle  that  will 
pass  through  the  points  A,  B,  C,  and  that  will  be  the  place  of 
the  church ;  the  measure  of  which,  to  any  of  these  points,  is 
the  answer  for  the  distance  :  draw  a  line  from  any  of  the  three 
points  to  the  centre,  and  the  angle  it  makes  with  either  of  the 
sides  that  contain  the  angle  it  was  drawn  to  ;  that  angle  laid  off 
by  the  direction  of  an  instrument,  on  the  ground,  and  the  dis- 
tance before  found,  being  ranged  thereon,  will  give  the  place  of 
the  church  required. 

By  Calculation. 

1.  'AB  :  AC+BC  :  :  AC— EC  :  AD— DB. 
2.5  4.3  .3  .516. 

1.25-f.258=1.508=AD. 

By  cor.  2,  theo.  14,  geom.,  the  square  root  of  the  difference 
of  the  squares  of  the  hypothenuse  AC,  and  given  leg  AD,  will 
give  DC. 

That  is,  5.29—2.274064=3.015936. 


OF  DISTANCES.  149 

Its  square  root  is  1.736= CD.  . 

Then,  by  the  preceding  lemma, 

2.  CD  :  AC  :  :  CB  :  the  diameter. 

1.736  2.3          2  2.65. 

the  half  of  which,  viz.  1.325,  is  the  semidiameter,  or  distance 
of  the  church  from  each  seat,  that  is,  AE,  CE,  BE. 

From  the  centre  E  let  fall  a  perpendicular  upon  any  of  the 
sides  as  EF,  and  it  will  bisect  it  in  jP  (by  theo.  8,  geom.). 
Wherefore,  AF=CF=±AC=1.15. 

In  the  right-angled  triangle  AFE  you  have  AF  1.15,  and 
AE  the  radius  1.325  given,  to  find  FAE.     Thus  : 
3.  AF  :  R  :  :  AE  :  sec.  FAE. 
1.15  90°    1.325      29°  47'. 

Wherefore,  directing  an  instrument  to  make  an  angle  of  29°  47' 
with  the  line  AC,  and  measuring  1.325  on  that  line  of  direction, 
will  give  the  place  of  the  church,  or 'the  centre  of  a  circle  that 
will  pass  through  A,  JB,  and  C. 

The  above  angle  1£AE  mayv  be  had  ^hout  a  secant,  as  be- 
fore. *.Ttus  :"*  V-  »  >  ' 

"  AE : R : : AF : & 

1.325  90°     1.15     60°  13'. 
Its  complement  29°  47'  will  give  FAE,  as  before. 

PRACTICAL    QUESTIONS. 

Ex.  1.  At  170  feet  distance  from  the  bottom  of  a  tower  the 
angle  of  its  elevation  was  found  to  be  52°  30'.  Required  the 
altitude  of  the  tower.  Ans.  221.55  feet. 

Ex.  2.  From  the  top  of  a  tower,  by  the  seaside,  of  14.3 
feet  high,  it  was  observed  that  the  angle  of  depression  of  a 
ship's  bottom,  then  at  anchor,  measured  35°.  What  then  was 
the  ship's  distance  from  the  bottom  of  the  wall  ? 

Ans.  204.22  feet. 

Ex.  3.  From  a  window  near  the  bottom  of  a  house  which 
seemed  to  be  on  a  level  with  the  bottom  of  a  steeple,  I  took  the 
angle  of  elevation  of  the  top  of  the  steeple,  equal  40°  ;  then 
from  another  window,  18  feet  directly  above  the  former,  the  like 
angle  was  37°  30'.  What  then  is  the  height  and  distance 
the  steeple  ?  An  J  height  **&6. 

3*  {  distance  MP9.50. 

Ex.  4.     Wanting  to  know  the  height  of  an  inaccessible  tower 
at  the  least  distance  from  it,  on  the  same  horizontal  plane,  I 


150  OF  DISTANCES. 

took  its  angle  of  elevation,  equal  to  58° ;  then  going  300  feet 
directly  from  it,  found  the  angle  there  to  be  only  32°.  Re- 
quired its  height  and  my  distance  from  it  at  the  first  station. 

Ang   (height    307.53. 

»  '  \  distance  192. 15. 

Ex.  5.  Being  on  the  side  of  a  river,  and  wanting  to  know 
the  distance  to  a  house  which  was  seen  on  the  other  side,  I 
measured  out  for  a  base  400  yards  in  a  right  line  by  the  side  of 
the  river,  and  found  that  the  two  angles,  one  at  each  end  of  this 
line,  subtended  by  the  other  end  and  the  house,  were  68°  2'  and 
73°  15'.  What  then  was  the  distance  between  each  station 
and  the  house  ?  .  (  593.08 

S<  \  612.38 

Ex.  6.  Wanting  to  know  the  breadth  of  a  river,  I  measured 
a  base  of  500  yards  in  a  straight  line  close  by  one  side  of  it, 
and  at  each  end  of  this  line  I  found  the  angles  subtended  by 
the  other  end  and  a  tree  close  to  the  bank  on  the  other  side  of 
the  river  to  be  53°  and  73°  15'.  What  then  was  the  perpen- 
dicular breadth  of  rife  river?  .^^/^-4t^Ans.  &t&. 4&ry aids. 

Ex.  7.  Two  ships  of  war,  intending  to  cann1fna(fe  Jrfort,  are, 
by  the  shallowness  of  the  water,  kept  so  far  from  it  that  they 
suspect  their  guns  cannot  reach  it  with  effect.  In  order  there- 
fore to  measure  the  distance,  they  separate  from  each  other  a 
quarter  of  a  mile,  or  440  yards  ;  then  each  ship  observes  and 
measures  the  angle  which  the  other  ship  and  the  fort  subtend, 
which  angles  are  83°  45'  and  85°  15'.  What  then  is  the  dis- 
tance between  each  ship  and  the  fort  ? 


Ex.  8.  A  point  of  land  was  observed  by  a  ship  at  sea  to 
bear  east  by  south  ;  and  after  sailing  north-east  12  miles,  it  was 
found  to  bear  south-east  by  east.  It  is  required  to  determine 
the  place  of  that  headland,  and  the  ship's  distance  from  it  at 
the  last  observation.  Ans.  26.0728  miles. 

Ex.  9.  If  the  height  of  the  mountain  called  the  Peak  of 
Teneriffe  be  21  miles,  as  it  is  very  nearly,  and  the  angle  taken  at 
the  top  of  it,  as  formed  between  a  plumb-line  and  a  line  conceived 
to  touch  the  earth  in  the  horizon,  or  farthest  visible  point,  be  88° 
2' ;  it  is  required  from  these  measures  to  determine  the  magni- 
tude^f  the  whole  earth,  and  the  utmost  distance  that  can  be 
seep  cm.  its  surface  from  the  top  of  the  mountain,  supposing  the 
form  of  the  earth  to  be  perfectly  globular. 

.        (  distance  135.943  >     -, 
Ans'  \  diameter      7916  \  miles' 


TO  FIND  THE  CONTENTS  OF  GROUND.     151 

Ex.  10.  Wanting  to  know  the  extent  of  a  piece  of  water, 
or  distance  between  two  headlands,  I  measured  from  each  of 
them  to  a  certain  point  inland,  and  found  the  two  distances  to  be 
735  yards  and  840  yards  ;  also  the  horizontal  angle  subtended 
between  these  two  lines  was  55°  40'.  What  then  was  the  dis- 
tance required?  Ans.  741.2  yards. 

Ex.  11.  Wanting  to  know  the  distance  between  a  house 
and  a  mill  which  were  seen  at  a  distance  on  the  other  side  of 
a  river,  I  measured  a  base  line  along  the  side  where  I  was  of 
600  yards,  and  at  each  end  of  it  took  the  angles  subtended  by 
the  other  end  and  the  house  and  mill,  which  were  as  follows, 
viz.  at  one  end  the  angles  were  58°  20'  and  95°  20',  and  at  the 
other  end  the  like  angles  were  53°  30'  and  98°  45'.  What 
then  was  the  distance  between  the  house  and  mill  ? 

Ans.  962.5866  yards. 

Ex.  12.*  Wanting  to  know  my  distance  from  an  inaccessible 
object  0  on  the  other  side  of  a  river,  and  having  no  instrument 
for  taking  angles,  but  only  a  chain  or  cord  for  measuring  dis- 
tances ;  from  each  of  two  stations,  A  and  .B,  which  were  taken 
at  500  yards  asunder,  I  measured,  in  a  direct  line  from  the  ob- 
ject 0,  100  yards,  viz.  AC  and  BD,  each  equal  100  yards; 
also  the  diagonal  AD  measured  550  yards,  and  the  diagonal 
BC  560.  What  then  was  the  distance  of  the  object  0  from 
each  station  A  and  B1  A  i  A  0  526.81. 

An8'  \B  0  500.47. 


SECTION  III. 
MENSURATION  OF  AREAS, 

OR    THE    VARIOUS    METHODS   OF  CALCULATING  THE    SUPERFICIAL 
CONTENTS    OF    ANY  FIELD. 

Definition. 

THE  area  or  contents  of  any  plane  surface  in  perches  is  the 
number  of  square  perches  which  that  surface  contains. 

*  These  practical  examples  are  taken  from  Button's  Mathematics,  vol. 
ii.  seventh  London  edition. 


152  TO  FIND  THE  CONTENTS  OF  GROUND. 

PL.  7.  Jig.  1. 

Let  A  BCD  represent  a  rectangular  parallelogram,  or  oblong ; 
let  the  side  AB  or  DC  contain  eight  equal  parts,  and  the  side  AD 
or  BC  three  of  such  parts ;  let  the  line  AB  be  moved  in  the 
direction  of  AD  till  it  has  come  to  EF,  where  AE  or  BF  (the 
distance  of  it  from  its  first  situation)  may  be  equal  to  one  of 
the  equal  parts.  Here  it  is  evident  that  the  generated  oblong 
ABEF  will  contain  as  many  squares  as  the  side  AB  contains 
equal  parts,  which  are  eight ;  each  square  having  for  its  side  one 
of  the  equal  parts  into  which  A  B  or  AD  is  divided.  Again,  let 
AB  move  on  till  it  comes  to  GH,  so  as  GE  or  HF  may  be 
equal  to  AE  or  BF;  then  it  is  plain  that  the  oblong  AGHB 
will  contain  twice  as  many  squares  as  the  side  AB  contains 
equal  parts.  After  the  same  manner  it  will  appear  that  the 
oblong  ADCB  will  contain  three  times  as  many  squares  as  the 
side  AB  contains  equal  parts  ;  and,  in  general,  that  every 
rectangular  parallelogram,  whether  square  or  oblong,  contains 
as  many  squares  as  the  product  of  the  number  of  equal  parts  in 
the  base  multiplied  into  the  number  of  the  same  equal  parts  in 
the  height  contains  units,  each  square  having  for  its  side  one 
of  the  equal  parts. 

Hence  arises  the  solution  of  the  following  problems. 


PROBLEM  I. 

To  find  the  contents  of  a  square  piece  of  ground. 

1.  Multiply  the  base  in  perches  into  the  perpendicular  in 
perches,  the  product  will  be  the  contents  in  perches ;  and  be- 
cause 160  perches  make  an  acre,  it  must  thence  follow  that 

Any  area,  or  contents  in  perches,  being  divided  by  160,  will 
give  the  contents  in  acres  ;  the  remaining  perches,  if  more  than 
40,  being  divided  by  40,  will  give  the  roods,  and  the  last  re 
mainder,  if  any,  will  be  perches, 

Or  thus : 

2.  Square  the  side  in  four-pole  chains  and  links,  and  the 
product  will  be  square  four-pole  chains  and  links  :  divide  this 
by  10,  or  cut  off  one  more  than  the  decimals,  which  are  five  in 
all,  from  the  right  towards  the  left :  the  figures  on  the  left  are 
acres ;  because  10  square  four-pole  chains  make  an  acre,  and 
the  remaining  figures  on  the  right  are  decimal  parts  of  an  acre. 
Multiply  the  five  figures  to  the  right  by  four,  cutting  five  figures 


TO  FIND   THE  CONTENTS  OF  GROUND.    153 

from  the  product,  and  if  any  figure  be  to  the  left  of  them  it 
is  a  rood,  or  roods ;  multiply  the  last  cut  off  figures  by  40, 
cutting  off  five,  or  (which  is  the  same  thing)  by  4,  cutting 
off  four;  and  the  remaining  figures  to  the  left,  if  any,  are 
perches. 

1.  The  first  part  is  plain,  from  considering  that  a  piece  of 
ground  in  a  square  form,  whose  side  is  a  perch,  must  contain  a 
perch  of  ground ;  and  that  40  such  perches  make  a  rood,  and 
four  roods  an  acre ;  or,  which  is  the  same  thing,  that  160  square 
perches  make  an  acre,  as  before. 

2.  A  square  four-pole  chain  (that  is,  a  piece  of  ground  four 
poles  or  perches  every  way)  must  contain  160  square  perches ; 
and  160  perches  make  an  acre ;  therefore  10  times  16  perches, 
or  10  square  four-pole  chains,  make  an  acre. 

Note. — The  chains  given  or  required,  in  any  of  the  following 

problems,  are  supposed  to  be  two-pole  chains,  that  chain  being 

most  commonly  used ;  but  they  must  be  reduced  to  four-pole 

.  chains  or  perches  for  calculation,  because  the  links  will  not 

operate  with  them  as  decimals. 


EXAMPLES. 
PL.  I.  Jig.  17. 

Let  ABCD  be  a  square  field,  whose  side  is  14cA.  291. ;  re- 
quired the  contents  in  acres. 

By  problem  4,  section  1,  part  2,  14cA.  292.  are  equal  to 

29.16  perches 

29.16 

17496 

2916 
26244 
5832 

A.  R.  P. 


160)850.3056(     5   1    10,  contents. 
40)50(1  rood. 
10  perches. 

Or  thus: 

14cA.  29Z.  are  equal  to  7ch.  29Z.  of  four-pole  chains,  by  prob- 
lem 1,  section  1,  part  2. 

63 


154  TO  FIND  THE  CONTENTS  OF  GROUND. 

eh.  I 
7.29 
7.29 

6561 
1458 
5103 

A.  R.  P. 

Acres  5|31441  contents/  as  before,   5    1  10. 
4 

Rood  1  [25764 
40 

Perches  10J30560 

It  is  required  to  lay  down  a  map  of  this  piece  of  ground,  by 
a  scale  of  twenty  perches  to  an  inch. 

Take  29.16,  the  perches  of  the  given  side,  from  the  small 
diagonal  on  the  common  surveying  scale,  where  twenty  small, 
or  two  of  the  large  divisions  are  an  inch :  make  a  square  whose 
side  is  that  length  (by  prob.  9,  geoin.),  and  it  is  done. 

PROBLEM  II. 

To  find  the  side  of  a  square  whose  contents  are  given. 

Extract  the  square  root  of  the  given  contents  in  perches,  and 
you  have  the  side  in  perches,  and  consequently  in  chains. 

EXAMPLE. 

It  is  required  to  lay  out  a  square  piece  of  ground  which  shall 
contain  12A.  3R.  16P.  Required  the  number  of  chains  in  each 
side  of  the  square  ;  and  to  lay  down  a  map  of  it  by  a  scale  of 
40  perches  to  an  inch. 

A.  R.  P. 
12   3  16 
4 

51 
40 

2056 


TO  FIND  THE  CONTENTS  OF  GROUND.     155 

2056(45.34+  perches  =22c/i.33|/.byprob.  6. 

sect  1,  part  2. 
85)456 

«  903)3100 
9064)39100  &c. 

To  draw  the  map. 

From  a  scale  where  4  of  the  large  or  40  of  the  small  divi- 
sions are  an  inch,  take  45.34  the  perches  of  the  side,  of  which 
make  a  square. 

PROBLEM  m. 

To  find  the  contents  of  an  oblong  piece  of  ground, 
Multiply  the  length  by  the  breadth,  for  the  contents. 

EXAMPLE. 

PL.  I.  Jig.  3. 

Let  ABCDbe  an  oblong  piece  of  ground,  whose  length^.1?  is 
I4tch.  251.  and  breadth  Sch.  371.  Required  the  contents  in  acres, 
and  also  to  lay  down  a  map  of  it,  by  a  scale  of  20  perches  to 
an  inch. 

ch.  I.  perches. 


15732 
3496 
A.   R.   P. 

160)506.9200(3     0    27  contents. 

26  perches,  or  near  27. 

Or  thus: 
four-pole  ch. 
ch.  I    ch.  I 

B?  Prob'  X»  sect' 


5075 
2175 
2900 

31682 


156  TO  FIND  THE  CONTENTS  OF  GROUND. 

Acres  3(16825 
4 

Rood  |67300 

'    •  4 

Perches  26|9200       . 

To  draw  the  map. 

Make  an  oblong  (by  schol.  to  prob.  9,  geom.)  whose  length, 
from  a  scale  of  20  to  an  inch,  may  be  29  perches,  and  breadth 
17.48  perches. 

PROBLEM  IV. 

The  contents  of  cm  oblong  piece  of  ground  and  one  side  given,  to  find  the  other* 

Divide  the  contents  in  perches  by  the  given  side  in  perches, 
the  quotient  is  the  side  required  hi  p.erches  ;  and  thence  it  may 
be  easily  reduced  to  chains. 

EXAMPLE. 

There  is  a  ditch  14cA.  251.  long,  by  the  side  of  which  it  is 
required  to  lay  out  an  oblong  piece  of  ground  which  shall  con- 
tain 3A.  OR.  27P.  What  breadth  must  be  laid  off  at  each  end 
of  the  ditch  to  enclose  the  3A.  OR.  27P.  ? 

A.  R.  P. 
3  0  27 
4 

12 

40 

perch.  cTi.  I. 

29)507(17.48=8  37,  breadth. 

217 

140 



240 

8 
The  map  is  constructed  like  the  last. 


TO  FIND  THE  CONTENTS  OF  GROUND.     1&7 


PROBLEM  V. 

To  find  the  contents  of  apiece  of  ground  in  form  of  an  ollique  angular  par- 
ttttlogram)  or  of  a  rhombus  or  rhomboidet. 

RULE    I. 

Multiply  the  base  into  the  perpendicular  height.    The  reason 
is  plain  from  theo.  13,  geom. 

EXAMPLE. 


Let  ABCD  be  a  piece  of  ground  in  form  of  a  rhombus,  whose 
base  AB  is  22  chains,  and  perpendicular  DE  or  FC  20  chains. 
Required  the  contents. 
ch. 
22  ^l  10 


Acres  11|0 
ch.  Or, 


160)1760(11  acres. 
160 


The  converse  of  this  is  done  by  prob.  4,  and  the  map  is  drawn 
by  laying  off  the  perpendicular  on  that  part  of  the  base  from 
whence  it  was  taken,  joining  the  extremity  thereof  to  that  of 
the  base  by  a  right  line,  and  thence  completing  the  parallelo- 
gram. 

RULE  n. 

As  rad.  (viz.  S.  of  90°,  pr  tang,  of  45°) 
Is  to  the  sine  of  any  angle  of  a  parallelogram, 
So  is  the  product  of  the  sides  including  the  angle  : 
To  the  area  of  the  parallelogram. 

That  is,  DA  x  AB  x  nat  sine  of  the  angle  A  =  the  area.* 
PI.  7,  fig.  2. 

*  Demonstration.  For,  having  drawn  the  perpendicular  DE  the  area 
by  the  first  rule  L*  ABxDE ;  but  as  radius  1  (S.  L.  E)  :  S.  L.  A  :  :  AD  : 


r 

158  TO  FIND  THE  CONTENTS  OF  GROUND. 

EXAMPLE. 

How  many  acres  are  in  a  rhomboides  whose  less  angle  is 
30°  and  the  including  sides  25.35  and  10.4  four-pole  chains  ? 

Ans.  13A.  OR.  29.12P. 

(Rad.)  1  :  .500000  (Nat.  S.  of  30°)  :  :  263.640  ( =25.35  X 
10.4)  :  131.82  =  the  area  in  four-pole  chains ;  which  divided 
by  10  (because  10  square  chains  are  an  acre)  gives  13.182 
acres,  or,  13 A.  OR.  29.12P. 

Note. — Because  the  angle  of  a  square  and  rectangle  are 
each  90°,  whose  sine  is  1,  this  rule  for  them  is  the  same  as  the 
first. 

PROBLEM  VI. 

To  find  the  contents  of  a  triangular  piece  of  ground. 

Multiply  the  base  by  half  the  perpendicular,  or  the  perpendicu- 
lar by  half  the  base  ;  or  take  half  the  product  of  the  base  into 
the  perpendicular.  • 

The  reason  of  this  is  plain  from  cor.  2,  theo.  12,  geom. 

EXAMPLE. 

PL.  1.  jig.  16. 

Let  ABC  be  a  triangular  piece  of  ground  whose  longest 
side  or  base  BC  is  24cA.  38/.,  and  perpendicular  AD^  let  fall 
from  the  opposite  angle,  is  13cA.  28?.  Required  the  contents. 

ch.L.     ch.l 
1.    Base  24.38  =  12.38  ).          ,      ,    . 

)     3  39  S  four'P°le  cnams> 


Acres  4]  19682 
4 

Rood    178728 
40 

Perches  31|49120 
Contents,  4 A.  OR.  3 IP. 


DE=S.  £-Ax  DA ;  therefore,  DE  xAB=ABx  S./LAx DA  =  the  area, 
or,  1  :  S.L.A  :  :  DAxAB :  S.l-AxDAxAB  =  the  area  oftheparal 
lelogram.  .Q.  E.  D. 


TO  FIND  THE  CONTENTS  OF  GROUND.    159 

ch.  I     ch.l 


Or,2dly.     Perp.    6.78  of  four-pole  chains. 
base  6.19 


6102 
678 
4068 

4|19682=4A.  OR.  31P. 

ch.  I 

Or,  3dly.     Base  12.38  four-pole  chains. 
Perp.    6.78 


9904 
8666 

7428 

83.9364 
Its  half  =4J19682=4A.  OR.  31P. 

Or  the  base  and  perpendicular  may  be  reduced  to  perches, 
and  the  contents  may  thence  be' obtained,  thus  : 


ch.  I.    perches. 

By  prob.  4,  sect.  1,  part  2. 


Perp.  13.28=27.12} 


Half  the  perp.  13.563 


perches,  ch.  Z. 
1.  Base     49.52=24.38 
i  perp.  13.56 

29712 
24760 
14856 
4952 


160)671.4912(4A.  OR.  31P. 
31 


160  TO  FIND  THE  CONTENTS  OF  GROUND. 

perches. 

2.  Perp.         27.12 
Half-base  24.76 

16272 
18984 
10848 
5424 


671.4912=4A.  OR.  31P. 

But  square  perches  may  be  reduced  to  acres,  &c.  rather 
more  commodiously  by  dividing  by  40  and  4,  than  by  160; 
thus, 

4|0)67|1 

4)16.  31 




A.  4.  0.  31 

*  -  


perches. 

3.    Base  49.52 

Perp.  27. 12 

9904 
.4952 
34664 
9904 


1342.9824 
671.4912=4A.OR.  31P. 

The  map  may  be  readily  drawn,  having  the  distance  from 
either  end  of  the  base  to  the  perpendicular  given ;  as  may  be 
evident  from  the  figure. 

PROBLEM  VII. 

The  contents  of  a  triangular  piece  of  ground  and  the  base  given,  to  find 
the  perpendicular. 

Divide  the  contents  in  perches  by  half  the  base  in  perches, 
and  the  quotient  will  give  you  the  perpendicular  in  perches,  and1 
so  in  chains. 


TO  FIND  THE  CONTENTS  OF  GROUND.     161 

EXAMPLE. 
PL.  I.  fig.  16. 

Let  BC  be  a  dilch,  whose  length  is  24ch.  40J.,  by  which  it  is 
required  to  lay  out  a  triangular  piece  of  ground,  whose  contents 
shall  be  4 A.  1R.  10P.  Required  the  perpendicular. 

ch.  I.  Perches. 
Base     24.40=49.6 
Half  the  base=24.8|v 


J7 

40 

Perches. 

24.8)690(27.82 


1940 


64 

Perches,  ch.  I. 

Answer,  perp.  27.82=13.45 

This  perpendicular  being  laid  on  any  part  of  the  base,  and 
lines  run  from  its  extremity  to  the  ends  of  the  base,  will  lay  out 
the  triangle  (by  cor.  to  theo.  13,  geom.)  so  that  the  perpen- 
dicular may  be  set  on  that  part  of  the  base  which  is  most  con- 
venient and  agreeable  to  the  parties  concerned. 

PRACTICAL    QUESTIONS. 

Ex.  1.  What  is  the  area  of  a  parallelogram  whose  length 
is  12.25  and  its  height  8.5  four-pole  chains  ? 

Ans.  10A.  1R.  26P. 

Ex.  2.  What  is  the  area  of  a  square  whose  side  is  70.25 
two-pole  chains?  Ans.  124A.  1R.  IP. 

Ex.  3.  What  is  tne  area  of  a  rhombus  whose  side  is  60 
perches,  and  its  height  45  perches  ?  Ans.  16 A.  3R.  SOP. 


162  TO  FIND  THE  CONTENTS  OF  GROUND. 

Ex.  4.  What  is  the  area  of  a  rhomboides  whose  less  angle 
is  40°  and  the  including  sides  80  and  25  four-pole  chains  ? 

Ans.  128 A.  2R.  9P. 

Ex.  5.  What  is  the  area  of  a  triangle  whose  base  is  12 
and  its  perpendicular  height  6  two-pole  chains  ? 

Ans.  1A.  3R.  8P. 

LEMMA. 
PL.  8.  fig.  9. 

If  from  half  the  sum  of  the  sides  of  any  plane  triangle  ABC  each  particular 
side  be  taken,  and  if  the  half-sum  and  the  three  remainders  be  multiplied 
continually  into  each  other,  the  square  root  of  this  product  will  be  the  area, 
of  the  triangle. 

Bisect  any  two  of  the  angles,  as  A  and  B,  with  the  lines 
AD,  BD,  meeting  in  D ;  draw  the  perpendiculars  DE,  DF,  DG. 

The  triangle  AFD  is  equiangular  to  AED;  for  the  angle 
FAD=EAD  by  construction,  and  AFD=AED,  being  each  a 
right  angle,  and  of  consequence  ADF^ADE ;  wherefore  AD  : 
DE  :  :  AD  :  DF;  and  since  AD  bears  the  same  proportion 
to  DF  that  it  does  to  DE,  DF=DE,  and  the  triangle  AFD= 
AED.  The  same  way  DE=DG,  and  the  triangle  DEB= 
DGB,  and  FD=DE=DG;  therefore  Dwill  be  the  centre  of 
a  circle  that  will  pass  through  E,  F,  G. 

In  the  same  way,  if  A  and  C  were  bisected,  the  same  point 
D  would  be  had  ;  therefore  a  line  from  J)  to  C  will  bisect  C, 
and  thus  the  triangles  DFC,  DGC  will  be  also  equal. 

Produce  CA  to  H,  till  AH=EB  or  GB ;  so  will  HC  be 
equal  to  half  the  sum  of  the  sides,  viz.  to  ±AB-\-^AC-\-±BC ; 
for  FC,  FA,  EB  are  severally  equal  to  CG,  AE,  BG ;  and 
all  these  together  are  equal  to  the  sum  of  the  sides  of  the  tri- 
angle ;  therefore  FC+FA+EB  or  CH  are  equal  to  half  the 
sum  of  the  sides. 

FC=CH—AB,  for  AF=AE,  and  HA=EB-,  therefore 
HF=AB,  and  AF=CH—BC;  for  CF=CG,  and  AH= 
GB  ;  therefore  BC=HA+FC,  and  AC=CH—AH. 

Continue  DC  till  it  meets  a  perpendicular  drawn  upon  H  in 
K ;  and  from  K  draw  the  perpendicular  KI,  and  join  AK. 

Because  the  angles  AHK  and  AIK  are  two  right  ones,  the 
angles  HAl  and  K  together  are  equal  to  two  right ;  since  the 
angles  of  the  two  triangles  contain  four  right :  in  the  same  way 
FDE+FAE=  (two  right  angles  =)  FAE+IAH ;  let  FAE 
be  taken  from  both,  then  FDE==IAH,  and  of  course  FAE= 
K',  the  quadrilateral  figures  AFDE  and  KHAI  are  therefore 
similar,  and  have  the  sides  about  the  equal  angles  proportional; 


TO  FIND  THE  CONTENTS  OF  GROUND.     163 

and  it  is  plain  the  triangles  CFD  and  CHK  are  also  propor- 
tional: hence, 

FD  :  HA  :  :  FA  :  HK 

FD:FC::HK:  HC. 

Wherefore,  by  multiplying  the  extremes  and  means  in  both, 
it  will  be  the  square  of  FDx HKx HC=FC xFAxHAx 
HK :  let  HK  be  taken  from  both,  and  multiply  each  side  by 
CH;  then  the  square  of  CH  X  by  the  square  of  FD=FCx 
FAxHAxCH. 

It  is  plain  by  the  foregoing  problem,  that  ±ABxDE  +{BC 
xDG+±ACxFD  =  the  area  of  the  triangle;  or  that  half 
the  sum  of  the  sides,  viz.  CH x  FD  ==  the  triangle ;  wherefore, 
the  square  of  CH  x  by  the  square  of  FD—FC  xFAx  HA  X 
CH,  that  is,  the  half-sum  multiplied  continually  into  the  dif- 
ferences between  the  half-sum  and  each  side  will  be  the  square 
of  the  area  of  the  triangle,  and  its  root  the  area.  Q.  E.  D. 

Cor  1.  If  all  the  sides  be  equal,  the  rule  will  become 
s/faXi  aX±  aX±a=%aaV3,  for  the  equilateral  triangle 
whose  side  is  a. 

Hence  the  following  problem  will  be  evident. 

PROBLEM  VIII. 

The  three  sides  of  a  plane  triangle  given,  to  find  the  area. 
RULE.* 

From  half  the  sum  of  the  three  sides  subtract  each  side 
severally;  take  the  logarithms  of  half  the  sum  and  three 
remainders,  and  half  their  total  will  be  the  logarithm  of  the 
area :  or,  take  the  square  root  of  the  continued  product  of  the 
half-sum  and  three  remainders  for  the  area. 

EXAMPLES. 

PL.  8.  Jig.  9. 

1.    In  the  triangle  ABC  are 

(AB=W.64\ 

Given  <  AC=  12.28  >  four-pole  chains ;  required  the  area 
(  CB=  9.00  ) 


Sum  31.92 


*  The  demonstrat;on  of  this  is  plain  from  the  foregoing  lemma,  and  the 
nature  of  logarithms. 


164  TO  FIND  THE  CONTENTS  OF  GROUND. 

Half-sum  15.96  Log.  1.203033 

C  5.32  0.725912 

Remainders  1 3.68  —  0.565848 

(6.96  —  0.842609 

2)3.337402 


Answer,  sqr.  ch.  46.63          log.          1.668701 
or,    4.663  acres. 

Or,  15.96X5.32X3.68X6.96=2174.71113216;  the  square 
root  of  which  is  46.63,  for  the  area,  as  before. 

2.  What  quantity  of  land  is  contained  in  a  triangle,  the  three 
sides  of  which  are  80,  120,  and  160  perches  respectively  ? 

Ans.  29A.  7P. 

3.  What  quantity  of  land  is  contained  in  a  triangle,  the  three 
sides  of  which  are  20,  30,  and  40  four-pole  chains  ? 

Ans.  29A.  7.579P. 

4.  How  many  acres  are  in  a  triangle,  whose  sides  are  49, 
50.25,  25.69  four-pole  chains?          Ans.  61A.  1R.  39.68P. 

fc 

PROBLEM  IX. 

Two  sides  of  a  plane  triangle  and  their  included  angle  given,  to  find  the  area. 
RULE.* 

To  the  log.  sine  of  the  given  angle  (or  of  its  supplement  to 
180°  if  obtuse)  add  the  logarithms  of  the  containing  sides; 
the  sum  less  radius  will  be  the  logarithm  of  the  double  area. 
Or,  As  radius 

Is  to  the  sine  of  any  angle  of  a  triangle, 

So  is  the  product  of  the  sides  including  the  angle : 

To  double  the  area  of  the  triangle. 

,    ABxACxNu.  S.  of  /-A/Tn  e 
That  is, (PL  5,  fig.  17) = the  area. 

*  Demonstration.  This  follows  from  rule  2,  prob.  5,  and  from  the  na- 
ture of  logarithms,  because  a  triangle  is  half  a  parallelogram  of  the  same 
base  and  height. 

Or  thus,  PL.  II,  fig.  3. 

Let  AH  be  perpendicular  to  AB  and  equal  to  AC,  and  HE,  FCG 
parallel  to  AB ;  then  making  AH  (—AC)  radius,  AF(=CD)  will  be  the 
sine  of  CAD,  and  the  parallelograms  ABEH (the  product  of  the  given  sides) 
and  ABGF(ihe  double  area  of  the  triangle),  having  the  same  base  AB,  are 
in  proportion  as  their  heights  AH,  AF ;  that  is,  as  radius  to  the  sine  of 
the  given  angle ;  which  proportion  gives  the  operation  as  in  the  rule  above* 


TO  FIND  THE  CONTENTS  OF  GROUND.     165 

EXAMPLES. 
PL.  5.  fig.  16. 

Suppose  two  sides  AB,  AC  of  a  triangular  lot  ABC  form 
an  angle  of  30  degrees,  and  measure  one  64  perches,  and  the 
other  40.5,  what  must  the  contents  be  ? 

Given  angle  30°      sine    9.698970 

Confides 


2)1296     log.     3.112605 
160)648(4A.  8P.,  Answer. 


8 

Or  thus  : 

.500000  sine  LA 
64  AB 


32.000000 

40.5  AC 


2)1296.0000000 
160)648 

4A.  8P. 

2.  Required  the  area  of  a  triangle,  two  sides  of  which  are 
49.2  and  40.8  perches,  and  their  contained  angle  1441  degrees. 

Ans.  3A.  2R.  22P. 

3.  What  quantity  of  ground  is  enclosed  in  an  equilateral 
triangle,  each  side  of  which  is  100  perches,  either  angle  being 
60  degrees  ?  Ans.  27A.  10P. 

PROBLEM  X. 

To  find  the  area  of  a  trapezoid,  viz.  a  figure  bounded  by  four  right  lines, 
two  of  which  are  parallel,  but  unequal. 

RULE.* 

Multiply  the  sum  of  the  parallel  sides  by  their  perpendicular 
distance,  and  take  half  the  product  for  the  area. 

*  Demonstration.     The  trapezoid  ABCD  (pi.  14,  fig.  8)  is  equivalent  to 
the  rectangle  contained  by  its  altitude  and  half  the  sum  of  the  parallel 


166  TO  FIND  THE  CONTENTS  OF  GROUND. 

EXAMPLES. 

1.  Required  the  area  of  a  trapezoid,  of  which  the  parallel 
sides  are,  respectively,  30  and  49  perches,  and  their  perpen- 
dicular distance  61.6. 

61.6 
0+49=79 

2)4866.4 
Answer,  2433.2=15A.  33.2P. 

PL.  9.  fig.  10. 

2.  In  the  trapezoid  ABCD  the  parallel  sides  are,  AD  20 
perches,  BC  32,  and  their  perpendicular  distance  AB  26 ; 
required  the  contents.  Ans.  4A.  36P. 

PROBLEM  XL 

To  find  the  contents  of  a  trapezium. 
RULE    I.* 

Multiply  the  diagonal,  or  line  joining  the  remotest  opposite 
angles,  by  the  sum  of  the  two  perpendiculars  falling  from  the 
other  angles  to  that  diagonal,  and  half  the  product  will  be 
the  area. 

sides  BC  and  AD.  For  draw  CE  parallel  to  AB  (prob.  8),  bisect  ED 
in  F,  and  draw  FG  parallel  to  AB,  meeting  the  production  of  .BC  in  G. 
Because  BC  is  equal  to  AE,  BC  and  AD  are  together  equal  to  AE  and 
AD,  or  to  twice  AE  with  ED,  or  to  twice  AE  and  twice  EF,  that  is,  to 
twice  AF ;  consequently,  AF=%(BC-\-AD).  Wherefore,  the  rectangle 
contained  by  the  altitude  of  the  trapezoid  and  half  the  sum  of  its  parallel 
sides  is  equivalent  to  the  rhomboid  BF :  but  the  rhomboid  EG  is  equiva- 
lent to  the  triangle  ECD  (theo.  12,  cor.  2) ;  add  to  each  the  rhomboid  BE, 
and  the  rhomboid  BFis  equivalent  to  the  trapezoid  ABCD. 

Note. — On  this  proposition  is  founded  the  method  of  offsets,  which 
enters  so  largely  into  the  practice  of  land  surveying.  In  measuring  a  field 
of  a  very  irregular  shape,  the  principal«points  only  are  connected  by  straight 
lines,  forming  sides  of  the  component  triangles,  and  the  distance  of  each 
remarkable  flexure  of  the  extreme  boundary  is  taken  from  these  rectilineal 
traces.  The  exterior  border  of  the  polygon  is  therefore  considered  as  a 
collection  of  trapezoids,  which  are  measured  by  multiplying  the  mean  of 
each  pair  of  offsets  or  perpendiculars  into  their  base  or  intermediate  dis- 
tance, which  is  one  of  the  other  sides,  because  the  parallel  sides  are  per- 
pendicular to  it. 

*  Demonstration.    For  the  trapezium  ABDC  =;  the  triangles  ABC-\- 


TO  FIND  THE  CONTENTS  OF  GROUND.     107 

EXAMPLE. 
PL.  7.  fig.  3. 

Let  ABCD  be  a  field  in  form  of  a  trapezium,  the  diagonal 
AC  64.4  perches,  the  perpendicular  Bb  13.6,  andZW  27.2;  re- 
quired the  contents. 

Diagonal =64.4  )»»,., 
13.64+27.2=40.8  J  Multiply. 

2)2627.52 

160)131 376(8A.  33fP.,  Answer. 
1280 

33£  perches. 

Note. — The  method  of  multiplying  together  the  half-sums  of 
the  opposite  sides  of  a  trapezium  for  the  contents  is  erroneous, 
and  the  more  so  the  more  oblique  its  angles  are. 

To  draw  the  map,  set  off  Ab  28  perches,  and  Ad  34.4,  and 
there  make  the  perpendiculars  to  their  proper  lengths,  and  join 
their  extremities  to  those  of  the  'diagonal. 

Note. — When  one  of  the  diagonals  and  the  four  sides  of  a 
trapezium  are  given,  it  is  divided  into  two  triangles  whose  sides 
are  given ;  the  area  of  each  triangle  may  be  found  (by  prob  8), 
and  their  sum  will  give  the  area  of  the  trapezium. 

RULE    II.* 

If  there  be  drawn  two  diagonals  cutting  each  other,  the  pro- 
duct of  the  diagonals  multiplied  by  the  natural  sine  of  the  angle 
of  intersection  of  the  diagonals  wilTbe  double  the  area.  And  this 
rule  is  common  to  a  square,  rhombus,  rhomboides,  &c.,  as  well  as 

to  all  other  quadrilateral  figures ;  that  is, — 

2 

=  the  area.     PI.  \4,  fig.  9.     Or,  as  radius  :S.LR::  ±ACX 
BD  :  the  area. 

Note. — Because  the  diagonals  of  a  square  and  rhombus  in- 
tersect at  a  right  angle,  whose  sine  is  1,  therefore  half  the  pro- 
duct of  their  diagonals  is  the  area. 

*  Pemonstration.  P1.14,fig.9.  For  the  trapez.  =  the  four  &sARB,BRC> 
CRD,  DRA=.(ARxRB-\-BR xRC+CRx RD-\-DRxRA) xlS.LR 
=(AR+RCxBR+CR-\-RAxDR)x1}S./L_R=AR+RCxBR-\-RD 
X±S.LR=ACxBDx]iS.<LR.  Q.  E.  D. 


168  TO  FIND  THE  CONTENTS  OF  GROUND. 

EXAMPLE. 

Let  the  two  diagonals  be  40  and  30  chains,  and  at  their  in- 
tersection one  of  the  less  angles  is  48°;  the  area  is  required. 

Then,  since  the  natural  sine  of  48°  is  .7431448,  the  area  = 
40  X  30  X.7431448  .  chains  = 


44A.  2R.  14.19P. 

By  Logarithms. 
Radius  10.000000 

Sine  of  48°  9.871073 

12*22=600          2.778151 

«* 


Area  445.8869       2.649224 

To  find  the  area  of  a  trapezium  when  three  side  sand  the  two  included 
angles  are  given. 

EXAMPLE. 

In  a  quadrangular  field  the  south  side  is  23.4,  the  east  side 
19.75,  and  the  north  side  20.5  chains  ;  also  the  south-east  and 
north-east  angles  are  73°  and  87°  30'.  What  is  the  area  ? 

First  (by  rule  2,  prob.  6)/9990482X219-75X2°-5=.4995241 

X  19.75  X  20.5=202.24482,  the  area  of  the  north-east  triangle 
BDC.     PL  14,  fig.  10. 

Again,  40.25  (=BC+CD)  :  .75  (=BC—  CD)  :  :  1.0446136 


Wherefore,  L.  #DC=46°  15'+!°  07'=47°  22';  and  LADB 
—  (ADC—  CJD5=73°—  47°  22'=)  25°  38'. 

But  L.BDC  47°  22'  9.8667026 
L.  C  87°  30'  9.9995865 
CB  20.5  1.3117539 


BD  1.4446378 
Whence  (by  rule  2,  prob.  6), 

\AD  11.7  1.0681859 

BD  1.4446378 

S.  L.ADB  25°  38'  9.6360969 


140.9031  the  area  of  the  A  ABD    2.1489206 


TO  FIND  THE  CONTENTS  OF  GROUND.     169 

Their  sum  is  343.1479  sq.  chains  =  34A.  1R.  and  10.3664P., 

the  area  required. 

• 

EXAMPLES    FOR    PRACTICE. 

1.  Required  the  area  of  a  trapezium  whose  diagonal  mea- 
sures 1 20  perches,  and  the  perpendiculars  24  perches  and  40 
perches.  Ans.  24  acres. 

2.  Required  the  area  of  a  trapezium  whose  diagonals  are  85 
and  24  four-pole  chains,  and  at  their  intersection  one  of  the  less 
angles  is  30°.  Ans.  105A. 

3.  What  is  the  area  of  a  trapezium  whose  south  side  is  27.4 
chains,  east  side  35.75  chains,  north  side  37.55  chains,  and 
west  side  4 1,05  chains ;  also  the  diagonal  from  south-west  to 
north-east  48.35  chains  ?  Ans.  123A.  1 L867P. 

PROBLEM  XII. 

To  find  the  area  of  a  circle  or  an  ellipsis. 
RULE. 

Multiply  the  square  of  the  circle's  diameter,  or  the  product 
of  the  longest  and  shortest  diameters  of  the  ellipsis,  by  .7854 
for  the  area.  Or,  subtract  0.104909  from  the  double  logarithm 
of  the  circle's  diameter,  or  from  the  sum  of  the  logarithms  of 
those  elliptic  diameters,  and  the  remainder  will  be  the  loga- 
rithm of  the  area. 

Note. — In  any  circle  the 

Diam.  multiplied  >  ,      Q  ,.  .-Q  i  produces  the  circum. 
Circum.  divided  ]  D  *.\  quotes  the  diam. 

EXAMPLES. 

1.   How  many  acres  are  in  a  circle  of  a  mile  diameter  ! 
1  mile  =  320  perches,  log.  2.505150 
2.505150 


5.010300 
0.104909 


4|0)8042J5     log.  4.905391 
4)2010.25 


Answer,  502 A.  2R.  25P. 
H 


170  TO  FIND  THE  CONTENTS  OF  GROUND. 

-$ 

2.  A  gentleman,  knowing  that  the  area  of  a  circle  is  greater 
than  that  of  any  other  figure  of  equal  perimeter,  walls  in  a  cir- 
cular deer-park  of  100  perches  diameter,  in  wjiich  he  makes 
an  elliptical  fish-pond  10  perches  long  by  5  wide.  Required 
the  length  of  his  wall,  contents  of  his  park,  and  area  of  his 
pond. 

Answer.  The  wall  314.16  perches,  enclosing  49A.  14P.,  of 
which  39|  perches,  or  £  of  an  acre  nearly,  is  appropriated  to 
the  pond. 

PROBLEM  XIII. 

The  area  of  a  circle  given,  to  find  its  diameter. 
RULE. 

To  the  logarithm  of  the  area  add  0.104909,  and  half  the  sum 
will  be  the  logarithm  of  the  diameter.  Or,  divide  the  area 
by  .7854,  and  the  square  root  of  the  quotient  will.be  the 
diameter. 

EXAMPLE. 

A  horse  in  the  midst  of  a  meadow  suppose 
Made  fast  to  a  stake  by  a  line  from  his  nose  : 
How  long  must  this  line  be  that,  feeding  all  round, 
Permits  him  to  graze 'just  an  acre  of  ground  ? 

Area  in  perches  160,  log.  2.204120 
0.104909 


2)2.309029 
Diameter 
2)14.2733     log.      1.154514 

Answer,  7.13665  per.=  117ft.  9in. 

PROBLEM  XIV. 

Allowance  for  roads. 

It  is  customary  to  deduct  6  acres  out  of  106  for  roads ;  the 
land  before  the  deduction  is  made  may  be  termed  ihetgross,  and 
that  remaining  after  such  deduction  the  neat. 

RULE. 

The  gross  divided    >  ^    j  06  $  (luotes  ^e  neat< 
The  neat  multiplied  $    ^          (  produces  the  g^oss. 


TO  FIND  THE  CONTENTS  OF  GROUND.     171 

EXAMPLES. 

1.  How  much  land  must  I  enclose  to  have  850 A.  2R.  20P. 
neat? 

4020 
4    2.5 

Acres.        A.   R.  P. 
850.625X1.06=901.6625=901  2  26,  the  answer. 

2.  How  much  neat  land  is  there  in  a  tract  of  901  A.  2R.  26P. 
gross  ? 


40 


26 

2.65 
Acres.       A.    R.  P. 


1.06)901.6625(850.625=850  2  20,  the  answer. 

848 

&c. 

Note. — These  two  operations  prove  each  other. 

PROBLEM  XV. 

To  find  the  area  of  apiece  of  ground ,  be  it  ever  so  irregular,  by  dividing 
it  into  triangles  and  trapezia. 

PL.  7.  /gr.4. 

We  here  admit  the  survey  to  be  taken  and  protracted ;  by- 
having,  therefore,  the  map,  and  knowing  the  scale  by  which  it 
was  laid  down,  the  contents  may  be  thus  obtained. 

Dispose  the  given  map  into  triangles  by  fine  pencilled  lines, 
such  as  are  here  represented  in  the  scheme,  and  number  the 
triangles  with  1 ,  2,  3,  4,  &c.  Your  map  being  thus  prepared, 
rule  a  table  with  four  columns,  the  first  of  which  is  for  the 
number  of  the  triangle,  the  second  for  the  base  of  it,  the 
third  for  the  perpendicular,  and  the  fourth  for  the  contents  in 
perches. 

Then  proceed  to  measure  the  base  of  number  1,  from  the 
scale  of  perches  the  map  was  laid  down  by,  and  place  that  in  the 
second  column  of  the  table,  under  the  word  base  ;  and  from  the 
angle  opposite  to  the  base  open  your  compasses  so  as  when  one 
foot  is  in  the  angular  point,  the  other,  being  moved  backward 
and  forward,  may  just  touch  the  base  line,  and  neither  go  the 
least  above  nor  beneath  it ;  that  distance  in  the  compasses, 
measured  from  the  §ame  scale,  is  the  length  of  that  perpendicu- 
lar, which  place  in  the  third  column  under  the  word  perpen- 
dicular. 

H2 


J72  TO  FIND  THE  CONTENTS  OF  GROUND. 

If  the  perpendiculars  of  two  triangles  fall  on  one  and  the  same 
base,  it  is  unnecessary  to  put  down  the  base  twice,  but  insert 
the  second  perpendicular  opposite  to  the  number  of  the  triangle 
in  the  table,  and  join  it  with  the  other  perp'endicular  by  a  brace, 
as  Nos.  1  and  2,  4  and  5,  6  and  7,  9  and  10,  &c. 

Proceed  after  this  manner  till  you  have  measured  all  the 
triangles,  and  then,  by  prob.  6,  find  the  contents  in  perches  of 
each  respective  triangle,  which  severally  place  in  the  table  op- 
posite to  the  number  of  the  triangle,  in  the  fourth  column,  under 
the  word  contents. 

But  where  two  perpendiculars  are  joined  together  in  the  table 
by  a  brace,  having  both  one  and  the  same  base,  find  the  con- 
tents of  each  (being  a  trapezium)  in  perches,  by  prob.  11,  which 
place  opposite  the  middle  of  those  perpendiculars,  in  the  fourth 
column,  under  the  word  contents. 

Having  thus  obtained  the  contents  of  each  respective  triangle 
and  trapezium  which  the  map  contains,  add  them  all  together, 
and  their  sum  will .  be  the  contents  of  the  map  in  perches, 
which  being  divided  by  1 60  gives  ifhe  contents  in  acres.  Thus,  for 

EXAMPLE. 


No. 

Base. 

Perpend. 

Contents. 

1 
2 

24.8 

17.0  ) 
16.35 

412.92 

3 

28.2 

16.0 

225.6 

4 
5 

39.8 

19.6) 
16.25 

712.42 

6 

7 

49.4 

29.0  > 
15.0  5 

1086.8 

8 

38.7 

6.7 

129.64 

9 
10 

40.0 

17.0  > 
13.05 

600 

11 
12 

42.8 

10.2) 
12.3  5 

481.5 

13 

26.2 

17.9 

234.49 

14 
15 

24.0 

11.6  > 
10.05 

259.2 

Contents  in  perches  

4142.57 

TO  FIND  THE  CONTENTS  OF  GROUND.     173 

This  being  divided  by  160  will  give  25A.  3R.  22P.,the  con- 
tents of  the  map. 

Let  your  map  be  laid  down  by  the  largest  scale  your  paper 
will  admit,  for  then  the  bases  and  perpendiculars  can  be  mea- 
sured with  greater  accuracy  than  when  laid  down  by  a  smaller 
scale,  and  if  possible  measure  from  scales  divided  diagonally. 

If  the  bases  and  perpendiculars  were  measured  by  four-pole 
chains,  the  contents  of  every  triangle,  and  trapezium  may  be  had 
as  before  in  problems  6  and  1 1 ,  and  consequently  the  whole 
contents  of  the  map. 

If  any  part  of  your  map  has  short  or  crooked  bounds,  as 
those  represented  in  plate  7,  fig.  5,  then  by  the  straight  edge  of 
a  transparent  horn  draw  a  fine  pencilled  line,  as  AB,  to  balance 
the  parts  taken  and  left  out,  as  also  another  BC :  these  parts, 
when  small,  may  be  balanced  very  nearly  by  the  eye,  or  they 
may  be  more  accurately  balanced  by  method  the  third.  Join 
the  points  A  and  C  by  a  line,  so  will  the  contents  of  the  triangle 
ABC  be  equal  to  that  contained  between  the  line  AC  and  the 
crooked  boundary  from  A  to  Bt  and  to  C :  by  this  method  the 
number  of  triangles  will  be  greatly  lessened,  and  the  contents 
become  more  certain ;  for  the  fewer  operations  you  have  the 
less  subject  will  you  be  to  err,  and  if  an  error  be  committed  the 
sooner  it  may  be  discovered. 

The  lines  of  the  map  should  be  drawn  small  and  neat,  as 
well  as  the  bases,  the  compasses  neatly  pointed,  and  the  scale 
accurately  divided  ;  without  all  which  you  may  err  greatly. 
The  multiplications  should  be  run  over  twice  at  least,  as  also 
the  addition  of  the  column  of  cpntents. 

From  what  has  been  said  it  will  be  easy  to  survey  a  field  by 
reducing  it  into  triangles  and  measuring  the  bases  and  perpen- 
diculars by  the  chain.  To  ascertain  the  contents  only  it  is  not 
material  to  know  at  what  part  of  the  base  the  perpendicular 
was  taken ;  since  it  has  been  shown  (in  cor.  to  theo.  1 3  geom.) 
that  triangles  on  the  same  base  and  between  the  same  paral- 
lels are  equal :  but  if  you  would  draw  a  map  from  the  bases 
and  perpendiculars,  it  is  evident  that  you  must  know  at  what 
part  of  the  base  the  perpendicular  was  taken,  in  order  to  set 
it  off  in  its  due  position ;  and  hence  the  map  is  easily  con- 
structed. 


174  TO  FIND  THE  CONTENTS  OF  GROUND. 


PROBLEM  XVI. 
PL.  8.  fig.  5. 

To  determine  the  area  of  a  piece  of  ground,  having  the  map  given,  oy 
reducing  it  to  one  triangle  equal  thereto,  and  thence  finding  its  contents. 

Let  ABCDEFGH  be  a  map  of  ground  which  you  would 
reduce  to  one  triangle  equal  thereto. 

Produce  any  line  of  the  map,  as  AH,  both  ways ;  lay  the 
edge  of  a  parallel  ruler  from  A  to  C,  having  B  above  it ;  hold 
the  other  side  of  the  ruler,  or  that  next  you,  fast ;  open  till  the 
same  edge  touches  B,  and  by  it,  with  a  protracting  pin,  mark 
the  point  b  on  the  produced  line  ;  lay  the  edge  of  the  ruler  from 
b  to  Z>,  having  C  above  it,  hold  the  other  side  fast,  open  till  the 
same  edge  touches  C,  and  by  it  mark  the  point  c  on  the  pro- 
duced line.  A  line  drawn  from  c  to  D  will  take  in  as  much  as 
it  leaves  out  of  the  map. 

Again,  lay  the  edge  of  the  ruler  from  H  to  F,  having  G  above 
it ;  keep  the  other  side  fast,  open  till  the  same  edge  touches  6r, 
and  by  it  mark  the  point  g  on  the  produced  line  ;  lay  the  edge 
of  the  ruler  from  g  to  JE,  having  jP  above  it,  keep  the  other  side 
fast,  open  till  the  same  edge  touches  F,  and  by  it  mark  the 
point /on  the  produced  line.  Lay  the  edge  of  the  ruler  from 
/to  J),  having  E  above  it,  keep  the  other  side  fast,  open  till 
the  same  edge  touches  E,  and  by  it  mark  the  point  e  on  the 
produced  line.  A  line  drawn  from  D  toe  will  take  in  as  much 
as  it  leaves  out.  Thus  have  you  the  triangle  cDe,  equal  to  the 
irregular  polygon  ABCDEFGH* 

If,  when  the  ruler's  edge  is  applied  to  the  points  A  and  C, 
the  point  B  falls  under  the  ruler,  hold  that  side  next  the  said 
points  fast,  and  draw  back  the  other  to  any  convenient  distance ; 
then  hold  this  last  side  fast,  and  draw  back  the  former  edge  to 
B,  and  by  it  mark  b  on  the  produced  line ;  and  thus  a  parallel 
may  be  drawn  to  any  point  under  the  ruler  as  well  as  if  it  were 
above  it.  It  is  best  to  keep  the  point  of  your  protracting  pin 
in  the  last  point  in  the  extended  line  till  you  lay  the  edge  of  the 
ruler  from  it  to  the  next  station,  or  you  may  mistake  one  point 
for  another. 

This  may  also  be  performed  with  a  scale  or  ruler  which  has 
a  thin-sloped  edge,  called  a  fiducial  edge,  and  a  fine-pointed 
pair  of  compasses.  Thus, 

Lay  that  edge  on  the  points  A  and  C;  take  the  distance  from 
the  point  B  to  the  edge  of  the  scale,  so  that  it  may  only  touch 
it,  in  the  same  manner  as  you  take  the  perpendicular  of  a  tri~ 
s* 

»  The  demonstration  of  this  is  evident  from  prob.  19,  Geom.,  page  63  of 
this  box>k. 


TO  FIND  THE  CONTENTS  OF  GROUND.     175 

angle ;  cany  that  distance  down  by  the  edge  of  the  scale  par- 
allel to  it  to  b,  and  there  describe  an  arc  on  the  point  6,  and  if 
it  just  touches  the  ruler's  edge  the  point  b  is  in  the  true  place 
of,  the  extended  line.  Lay  then  the  fiducial  edge  of  the  scale 
from  b  to  -D,  and  take  a  distance  from  C  that  will  just  touch  the 
edge  of  the  scale ;  cany  that  distance  along  the  edge  till  the 
point  which  was  in  C  cuts  the  produced "  line  in  c  j  keep  that 
foot  in  c  and  describe  an  arc,  and  if  it  just  touches  the  ruler's 
edge  the  point  c  is  in  the  true  place  of  the  extended  line.  Draw 
a  line  from  c  to  D  and  it  will  take  in  and  leave  out  equally :  in 
like  manner  the  other  side  of  the  figure  may  be  balanced  by  the 
line  eD. 

Let  the  point  of  your  compasses  be  kept  to  the  last  point  of 
the  extended  line  till  you  lay  your  scale  from  it  to  the  next 
station,  to  prevent  mistakes  from  the  number  of  points. 

That  the  triangle  cDe  is  equal  to  the  right-lined  figure 
ABCDEFGH  will  be  evident  from  problems  18,  19,  geom. ; 
for  thereby,  if  a  line  were  drawn  from  bfa  C,  it  will  give  and 
take  equally,  and  then  the  figure  bCDEFGH  will  be  equal  to 
the  map.  Thus  the  figure  is  lessened  by  one  side,  and  the 
next  balance  line  will  lessen  it  by  two,  and  so  on,  and  will  give 
and  take1  equally.  In  the  same  manner  an  equality  will  arise 
on  the  other  side. 

The  area  of  the  triangle  is  easily  obtained,  as  before,  and 
thus  you  have  the  area  of  the  map. 

It  is  best  to  extend  one  of  the  shortest  lines  of  the  polygon ; 
because  if  a  very  long  line  be  produced,  the  triangle  will  have 
one  angle  very  obtuse,  and  consequently  the  other  two  very 
acute ;  in  which  case  it  will  not  be  easy  to  determine  exactly 
the  length  of  the  longest  side,  or  the  points  where  the  balancing 
lines  cut  the  extended  one. 

This  method  will  be  found  very  useful  and  ready  in  small 
enclosures,  as  well  as  very  exact ;  it  may  be  also  used  in  large 
ones,  but  great  care  must  be  taken  of  the  points  on  the  extended 
line,  wliich  will  be  crowded,  as  well  as  of  not  missing  a  station. 


PROBLEM  XVII. 

A  map  with  its  area  being  given,  and  its  scale  omitted  *  It  either  draton  or 
mentioned,  to  find  the  scale. 

Cast  up  the  map  by  any  scale  whatsoever,  and  it  will  be 

As  the  area  found 

Is  to  the  square  of  the  scale  by  which  you  cast  np, 

So  is  the  given  area  of  the  map 

To  the  square  of  the  scale  by  which  it  was  laid  down. 

The  square  root  of  which  will  give  the  scale. 


176  TO  FIND  THE  CONTENTS  OF  GROUND. 


EXAMPLE. 

A  map  whose  area  is  126 A.  3R.  16P.  being  given,  and  the 
scale  omitted  to  be  either  drawn  or  mentioned,  to  find  the  scale. 

Suppose  this  map  was  cast  up  by  a  scale  of  20  perches  to 
an  inch,  and  the  contents  thereby  produced  be  31A.  2R.  34P. 

As  the  area  found,  31A.  2R.  34P.=5074P. 

Is  to  the  square  of  the  scale  by  which  it  was  cast  up,  that  is, 
to  20X20=400, 

So  is  the  given  area  of  the  map  126 A.  3R.  16P.=20296P. 

To  the  square  of  the  scale  by  which  it  was  laid  down. 

5074  :  400  : :  20296  :  1600,  the  square  of  the  required  scale. 

Root. 
1600(40 
16 

8)      00 

Answer.  The  map  was  laid  down  by  a  scale  of  40  perches 
to  an  inch. 

PROBLEM  XVIII. 

How  to  find  the  true  contents  of  a  survey,  though  it  be  taken  by  a  chain  that 
is  too  long  or  too  short. 

Let  the  map  be  constructed,  and  its  area  found,  as  if  the 
chain  were  of  the  true  length.     And  it  will  be, 
As  the  square  of  the  true  chain 
Is  to  the  contents  of  the  map, 
So  is  the  square  of  the  chain  you  surveyed  by 
To  the  true  contents  of  the  map. 

EXAMPLE. 

If  a  survey  be  taken  with  a  chain  which  is  3  inches  too  long, 
or  with  one  whose  length  is  42  feet  3  inches,  and  the  map 
thereof  be  found  to  contain  U20A.  2R.  20P. ;  required  the  true 
contents.  ^L 

As  the  square  of  42ft.  Om.  =  the  square  of  504  inches  = 
254016 

Is  to  the  contents  of  the  map,  920A.  1R.  20P.  — 147260P., 

So  is  the  square  of  42ft.  3t*9.  =  the  square  of  507  inches 
=  257049 

To  the  true  contents* 


COMPUTATION  OF  AREAS.  177 

P.  P. 

250416  :  147260  :  :  257049  :  149019 

A.  R.  P. 
160)149019(931  1  19,  Answer. 


METHOD  OF  DETERMINING  THE  AREAS  OF 
RIGHT-LINED  FIGURES  UNIVERSALLY,  OR  BY. 
CALCULATION. 

Definitions. 
PL.  8.  fig.  7. 

1.  MERIDIANS  are  north  and  south  lines,  which  are  sup- 
posed to  pass  through  every  station  of  the  survey. 

2.  The  difference  of  latitude,  or  the  northing  or  southing 
of  any  stationary  line,  is  the  distance  that  one  end  of  the  line 
is  north  or  south  from  the  other  end  ;  or,  it  is  the  distance  which 
is  intercepted  on  the  meridian,  between  the  beginning  of  the 
stationary  line  and  a  perpendicular  drawn  from  the  other  end  to 
that    meridian.      Thus,  if    NS  be  a  meridian  line  passing 
through  the  point  A  of  the  line  AB,  then  is  Ab  the  difference 
of  latitude  or  southing  of  that  line. 

3.  The  departure  of  any  stationary  line  is  the  nearest  dis- 
tance froimone  end  of  the  line  to  a  meridian  passing  through 
the  other  end.     Thus  Bb  is  the  departure  or  easting  of  the  line 
AB :  but  if  CB  be  a  meridian,  and  the  measure  of  the  sta- 
tionary distance  be  taken  from  B  to  A,  then  is  BC  the  differ- 
ence of  latitude,  or  northing,  and  .AC  the  departure  or  westing 
of  the  line  BA. 

4;  That  meridian  which  passes  through  the  first  station  is 
sometimes  called  the  first  meridian ;  and  sometimes  it  is  a  me- 
ridian passing  on  the  east  or  west  side  of  the  map,  at  the  dis- 
tance of  the  breadth  thereof,  from  east  to  west,  set  off  from  the- 
first  station. 

5.    The  meridian  distance  of  any  station  is  the  distance 


178  COMPUTATION  OF  AREAS. 

thereof  from  the  first  meridian,  whether  it  be  supposed  to  pass 
through  the  first  station  or  on  the  east  or  west  side  of  the  map. 

THEOREM  L 

In  every  survey  which  is  truly  taken,  the  sum  of  the  north- 
ings will  be  equal  to  that  of  the  southings  ;  and  the  sum  of  the 
eastings  equal  to  that  of  the  westings. 

PL.  9.  jig.  1. 

Let  abcefgh  represent  a  plot  or  parcel  of  land.  Let  a 
be  the  first~  station,  b  the  second,  c  the  third,  &c.  Let  NS" 
be  a  meridian  line ;  then  will  all  lines  parallel  thereto,  which 
pass  through  the  several  stations,  be  meridians  alscH;  as  a0,  bst 
cd,  &c.,  and  the  lines  bo,  cs,  de,  &c.,  perpendicular  to  those, 
will  be  the  east  or  west  lines  or  departures. 

The  northings,  ei+go-\-hq=ao+bs-\-cd-\-fr,l}\e  southings: 
for  let  the  figure  be  completed  ;  then  it  is  plain  that  go-\-hq+ 
rk=ao+bs+cd,  and  ei — rk=fr.  If  to  the  former  part,  of  this 
first  equation  ei — rk  be  added,  and/r  to  the  latter,  then  #0+ 
hq-\-ei=ao+bs-\-cd-\-fr,  that  is,  the  sum  of  the  northings  is 
equal  to  that  of  the  southings. 

'  The  eastings,  cs-{-qa— ob+de+if+rg+oh,  the  westings. 
For  aq-\-yo  (az)  —  de-\- if -\-rg-\-oh,  and  00— cs — yo.  If  to  the 
former  part  of  this  first  equation  cs — yo  be  added,  and  bo  to 
the  latter,  then  cs-\-aq=ob-^-de-\-if-{-rg-\-oh ;  that  is,  the  sum 
of  the  eastings  is  equal  to  that  of  the  westings.  Q.  E.  D. 

SCHOLIUM. 

This  theorem  is  of  use  to  prove  whether  the  field-work  be 
truly  taken  or  not ;  for  if  the  sum  of  the  northings  be  equal  to 
that  of  the  southings,  and  the  sum  of  the  eastings  to  that  of 
the  westings,  the  field-work  is  right,  otherwise  not; 

Since  the  proof  and  certainty  of  a  survey  depend  on  this 
truth,  it  will  be  necessary  to  show  how  the  difference  of  latitude 
and  departure  for  any  stationary  line,  whose  course  and  dis- 
tance are  given,  may  be  obtained  by  thea  table  usually  called 
the  Traverse  Table.* 

*  This  table  is  calculated  by  the  first  case  of  right-angled  plane  trigo- 
nometry, taught  in  the  fifth  section  of  the  first  part  of  this  book>  where- 
the  hypothenuse  and  an  acute  angle  are  given,  to  find  the  legs. 

In  the  right-angled  triangle  ABC  (PI.  8,  fig.  7),  given  the  distance  or 
hypothenuse  AB  91  chains,  links,  or  perches,  the  course  or  one  of  the 
acute  angles  ABC  41°  ;  it  is  required  to  find  the  legs,  or  the  difference  of 
latitude  and  departure.. 


m 

COMPUTATION  OF  AREAS.  179 

To  find  the  difference  of  latitude  and  departure  by  the  Traverse 
Table. 

This  table  is  so  contrived,  that  by  finding  therein  the  given 
course,  and  a  distance  not  exceeding  120  miles,  chains,  perches, 
or  feet,  the  difference  of  latitude  and  departure  is  had  by  in- 
spection :  the  course  is  to  be  found  at  the  top  of  the  table  when 
under  45  degrees,  but  at  the  bottom  of  the  table  when  above 
45  degrees.  Each  column  signed  with  a  course  consists  of  two 
parts,  one  for  the  difference  of  latitude,  marked  Lat.,  the  other 
for  the  departure,  marked  Dep.,  which  names  are  both  at  the 
top  and  bottom  of  these  columns.  The  distance  is  to  be  found 
in  the  column  marked  DisU,  next  the  left-hand  margin  of  the 


EXAMPLE. 

In  the  use  of  this  table,  a  few  observations  only  are  ne- 
cessary. 

1.  If  a  station  consist  of  any  number  of  even  chains  or 
perches  (which  are  almost   the  only  measures  used  in  survey- 
ing), the  latitude  and  departure  are  found   at  sight  under  tjae 
bearing  or  course,  if  less  than  45  degrees,  or  over  it  if  more, 
and  in  a  line  with  the  distance. 

2.  If  a  station  consist  of  any  number  of  chains  and  perches,. 
and  decimals  of  a  chain  or  perch,  under  the  distance  10,  the 
lat.  and  dep.  will  be  found  as  above,  either  over  or  under  the 
bearing  ;  the  decimal  point  or  separatrix  being  removed  one 
figure  to  the  left,  which  leaves  a  figure  to  the  right  to  spare. 

If  the  distance  be  any  number  of  chains  or  perches,  and  the 
decimals  of  a  chain  or  perch,  the  lat.  and  dep.  must  be  taken 


As  radius  90°          10.000000 

is  to  AB,  91  1.959041 

So  is  the  sine  of  B  41°  9.816943 

to  AC  59.70  1.775984 

As  radius  90°          10.000000 

is  to  315,  91  1.959041 

So  is  the  sine  of  A  49°  9.877780 

to  BC          68.68  1.836821 

Hence  AC  is  the  departure  and  BC  the  difference  of  latitude  which 
correspond  to  those  in  the  table.  In  the  same  manner  the  difference  of 
latitude  and  departure  to  every  degree  in  the  table  is  calculated,  by  which 
the  practitioner  can  at  any  time  proxe  the  exactness  of  those  in  the  table. 


180  COMPUTATION  OF  AREAS. 

out  at  two  or  more  operations,  by  taking  out  the  lat.  and  dep* 
for  the  chains  or  perches  in  the  first  place ;  and  then  for  the 
decimal  parts. 

To  save  the  repeated  trouble  of  additions,  a  judicious  sur- 
veyor will  always  limit  his  stations  to  whole  chains  or  perches 
and  lengths,  which  can  commonly  be  done  at  every  station 
save  the  last. 

1.  In  order  to  illustrate  the  foregoing  observations,  let  us 
suppose  a  course  or  bearing  to  be  S.  35°  15'  £.,  and  the  dis- 
tance 79  four-pole  chains.     Under  35°  15',  or  35|  degrees* 
and  opposite  79,  we  find  64.51  for  the  latitude,  and  45.59  the 
departure,  which  signify  that  the  end  of  that  station  differ  in 
latitude  from  the  beginning  64.51  chains,  and  in  departure  45.59 
chains. 

Note. — We  are  to  understand  the  same  things  if  the  distance 
is  given  in  perches  or  any  other  measures,  the  method  of  pro- 
ceeding being  exactly  the  same  in  every  case. 

Again,  let  the  bearing  be  54|  degrees,  and  distance  as  before ; 
then  over  said  degrees  we  find  the  same  numbers,  only  with 
this  difference,  that  the  lat.  before  found  will  now  be  the  dep., 
and  the  dep.  the  lat.,  because  54f  is  the  complement  of  35£  de- 
grees to  90,  viz.  lat.  45.59,  dep.  64.51. 

2.  Suppose  the  same  course,  but  the  distance  7  chains  90 
links,  or  as  many  perches.     Here  we  find  the  same  numbers, 
but  the  decimal  point  must  be  removed  one  figure  to  the  left. 

Thus,  under  35JL,  and  in  a  line  with  79  or  7.9,  are 

Lat.  6.45 

Dep.  4.56 

the  5  in  the  dep.  being  increased  by  1,  because  the  9  is  re- 
jected ;  but  over  54f  we  get 

Lat.  4.56 

Dep.  6.45 

3.  Let  the  course  be  as  before,  but  the  distance  7.79,  then 
opposite 

7.70  Lat.  6.29  Dep.  4.43 

976 

7.79  6.36  4.49 

Or  opposite 

7.00  Lat.  5.72  Dep.  4.03 

.79  .64  .4ft 

7.79  6.36  4.49- 


COMPUTATION  OF  AREAS.  181 

THEOREM  II. 

When  the  first  meridian  passes  through  the  map. 

If  the  east  meridian  distances  in  the  middle  of  each  line  be  multiplied  into 
the  particular  southing,  and  the  west  meridian  distances  into  the  particular 
northing)  the  sum  of  these  products  will  be  the  area  of  the  map. 

PL.  10.  fig.  1. 

Let  the  figure  abkm  be  a  map,  the  lines  ab,  Ik  to  the  south- 
ward, and  km,  ma  to  the  northward,  NS  the  first  meridian  line 
passing  through  the  first  station  a. 

The  meridian    (  zd  X  ao          > .        (  am 

distances  east    \tu  X  ox  (bq)    y  ~        ;a  (  ow 

The  meridian    <  efxgx  )  =  .        <  xp 

distances  west  \hhXga  (my)  $  '    \  gl 

These  four  areas  am+ow-\-  xp-\-gl  will  be  the  area  of  the 
whole  figure  cmswiprlc,  which  is  equal  to  the  area  of  the  map 
abkm.  Complete  the  figure. 

The  parallelograms  am  and  ow  are  made  of  the  east  meridian 
distances  dz  and  tu  multiplied  into  the  southings  ao  and  ox ;. 
the  parallelograms  xp  and  gl  are  composed  of  the  west  meri- 
dian distances  ef  and  hh  multiplied  into  the  northings  xg  and 
ga  (my) :  but  these  four  parallelograms  are  equal  to  the  area 
of  the  map  ;  for  if  from  them  be  taken  the  four  triangles  marked 
Z,  and  in  the  place  of  those  be  substituted  the  four  triangles 
marked  O,  which  are  equal  to  the  former,  then  it  is  plain 
the  area  of  the  map  will  be  equal  to  the  four  parallelograms. 
Q.  E.  D. 

THEOREM  III. 

If  the  meridian  distance  when  east  be  multiplied  into  the  southings,  and 
the  meridian  distance  when  west  be  multiplied  into  the  northings,  the  sum 
»f  these  less  by  the  meridian  distance  when  west  multiplied  into  the  south- 
ings  is  the  area  of  the  survey. 

PL.  10.  fig.  2. 

Let  abc  be  the  map. 

The  jgure  being  completed,  the  rectangle  af  is  made  of  the 
meridian  distance  eq  when  east  multiplied  into  the  southing 
an  ;  the  rectangle  yk  is  made  of  the  meridian  distance  xw,  mul- 
tiplied into  the  northings  cz  or  ya.  These  two  rectangles,  or 
parallelograms,  af-\-yk,  make  the  area  of  the  figure  dfnyikd ; 
from  which  taking  the  rectangle  oy,  made  of  the  meridian  dis- 
tance tu  when  west  into  the  southings  oh  or  bm,  the  remainder 


* 
182  COMPUTATION  OF  AREAS. 

is  the  area  of  the  figure  dfohikd,  which  is  equal  to  the  area  of 
the  map. 

Letbou  =  Y,  urih=L,  ric=O,  wrc=Z,  akw=Kf  cfb=B, 
and  ade=A.  I  say  that  Y+Z+B=K+L+A. 

F=Z+O;  add  Z  to  both,  then  Y+Z=L+O+Z:  but  Z 
+  O=K,  put  K  instead  of  Z+O,  then  Y+Z=L+K\  add 
to  both  sides  the  equal  triangles  B  and  A,  then  Y+Z+B=L 
+K+A.  If  therefore  B+Y+Zbe  taken  from  abc,  and  in 
lieu  thereof  we  put  L-\-K-\-  A,  we  shall  have  the  figure  dfoHikd 
=abc',  but  that  figure  is  made  up  of  the  meridian  distance  when 
east  multiplied  into  the  southing,  and  the  meridian  distance 
when  west  multiplied  into  the  northing  less  by  the  meridian 
distance  when  west  multiplied  into  the  southing.  Q.  E.  D. 

COROLLARY. 

Since  the  meridian  distance  when  west  multiplied  into  the 
southing  is  to  be  subtracted,  by  the  same  reasoning  the  me- 
ridian distance  when  east  multiplied  into  the  northing  must  be 
also  subtracted. 

SCHOLIUM. 

From  the  two  preceding  theorems  we  learn  how  to  find  the 
area  of  the  map  when  the  first  meridian  passes  through  it ; 
that  is,  when  one  part  of  the  map  lies  on  the  east  and  the  other 
on  the  west  side  of  that  meridian.  Thus, 

RULE. 

The  mend.  (  east  )  muhi  lied  into  the  <  southings, ) 
dist.  when   }  west  >  $  northings,  ) 

sum  is  the  area  of  the  map. 

But, 

TheineriiL  <  east  >     d  pliedintathe  $  ™*hings,  \  the  sum 
dist.  when  (  west  $  (  southings,  ^ 

of  these  products  taken  from  the  former  gives  the    area  of 
the  map. 

These  theorems  are  true  when  the  surveyor  keeps  the  land 
he  surveys  on  his  right-hand,  which  we  suppose  thrgugh  the 
whole  to  be  done ;  but  if  he  goes  the  contrary  way,  call  the 
southings  northings  and  the  northings  southings,  and  the  same 
rule  will  hold  good. 


COMPUTATION  OF  AREAS,  183 

General  Rule  for  finding  the  Meridian  Distances. 

1.  The  meridian  distance  and  departure  both  east  or  both 
west,  their  sum  is  the  meridian  distance  of  the  same  name. 

2.  The  meridian  distance  and  departure  of  different  names, 
that  is,  one  east  and  the  other  west,  their  difference  is  the  me- 
ridian distance  of  the  same  name  with  the  greater. 

Thus,  in  the  first  method  of  finding  the  area,  a&  in  the  follow- 
ing field-book, 

The  first  departure  is  put  opposite  the  northing  or  southing 
of  the  first  station,  and  is  the  first  meridian  distance  of  the  same 
name.  Thus,  if  the  first  departure  be  east,  the  first  meridian 
distance  will  be  the  same  as  the  departure,  and  east  also,,  and 
if  west  it  will  be  the  same  way. 

The  first  meridian  distance  6.61  E. 

The  next  departure  6.61  E, 

The  second  meridian  distance       13.22  E. 
The  next  departure  1.80  E. 


The  third  meridian  distance          15.02  E. 


At  station  5,  the  meridian  distance  5.78  E. 
The  next  departure  7.76  W 

The  next  meridian  distance  1.98  W. 


At  station  1 1,  the  meridian  distance  0. 12  W. 
The  next  departure  5.84  E. 

The  next  meridian  distance  5.72  E. 


PL.  10.  fig.  9. 

In  the  5th  and  llth  stations,  the  meridian  distance  being  less 
than  the  departures  and  of  a  contrary  name,  the  map  will 
cross  the  first  meridian,  and  will  pass,  as  in  the  5th  line,  from 
the  east  to  the  west  line  of  the  meridian ;  and  in  the  llth  line 
it  will  again  cross  from  the  west  to  the  east  side,  which  will 
evidently  appear  if  the  field-work  be  protracted,  and  the  me- 
ridian line  passing  through  the  first  station  be  drawn  through 
the  map. 

The  field-book  cast  up  by  the  first  method  will  be  evident 


184  COMPUTATION  OF  AREAS. 

from  the  two  foregoing  theorems^  and  therefore  requires  no- 
further  explanation ;  but  to  find  the  urea  by  the  second  method 
take  this 

RULE. 

When  the  meridian  distances  are  east,  put  the  products  of 
north  and  south  areas  in  their  proper  columns,  but  when  west 
in  their  contrary  columns  ;  that  is,  in  the  column  of  south  area 
when  the  difference  of  latitude  is  north,  and  in  north  when 
south :  the  reason  of  which  is  plain  from  the  last  two  theo- 
rems. The  difference  of  these  two  columns  will  be  the  area 
of  the  map. 

Construction  of  the  Map  from  either  the  first  or  the  second  Table. 
PL.  10.  Jig.  3. 

Draw  the  line  NS  for  a  north  and  south  line,  which  call  the 
first  meridian  ;  in  this  line  assume  any  point,  as  1,  for  the  first 
station.  Set  the  northing  of  that  stationary  line,  which  is  3.54, 
from  1  to  2,  on  the  said  meridian  line.  Upon  the  point  2 
raise  4  perpendicular  to  the  eastward,  the  meridian  distance 
being  easterly,  and  upon  it  set  13.22,  the  second  number  in  the 
column  of  meridian  distances  from  2  to  2,  and  draw  the  line 
1,  2  for  the  first  distance  line :  from  2  upon  the  first  meridian 
set  the  northing  of  the  second  stationary  line,  that  is,  9.65,  to  3, 
and  on  the  point  3  erect  a  perpendicular  eastward,  upon  which 
set  the  meridian  distance  of  the  second  station  16.82,  from  3  to 
3,  and  draw  the  line  2,  3,  for  the  distance  line  of  the  second 
station.  And  since  the  third  station  has  neither  northing  nor 
southing,  set  the  meridian  distance  of  it  33.02,  from  3  to  4,  for 
the*  distance  line  of  the  third  station.  To  the  fourth  station 
there  is  29.44  southing,  which  set  from  3  to  5  ^  upon  the  point 
5  erect  the  perpendicular  5,  5 ;  on  which  lay  13.54,  and  draw 
the  line  4  to  5. 

In  the  like  manner  proceed  to  set  the  northings  and  south- 
ings on  the  first  meridian,  and  the  meridian  distances  upon  the 
perpendiculars  raised  to  the  east  or  west ;  the  extremities  of 
which  connected  by  right  lines  will  complete  the  map. 


COMPUTATION  OF  AREAS. 

Field-book,  Method  I.      • 


185 


No. 

St. 

Bearings. 

C.L. 

Lat.  and 
halfDep. 

Mend. 
Dist. 

Area. 

Deduct. 

1 

NE  75° 

13.70 

N     3.54 
E     6.61 

6.61   E 
13.22  E 

23.3994 

o 

NE  20» 

10.30 

N     9.67 
E     1.80 

15.02  E 
16.82  E 

144.9430 

3 

East. 

16.20 

0.00 
E     8.10 

24.92  E 
33.02  E 

4 

SW  33i 

35.30 

S   29.44 
W    9.74 

23.28  E 
13.54  E 

685.3632 

5 

SW76 

_^_ 

16.00 

S      3.87 
W    7.76 

5.78  E 
1.98  W 

22.3686 

6 

•  North. 

9.00 

N     9.00 
0.00 

1.98  W 
1.98  W 

17.8200 

7 

SW84 

11.60 

S      1.21 
W    5.77 

7.75  W 
13.52  W 

9.3775 

8 

NW531 

11.60 

N     6.94 
W    4.64 

18.16  W 
22.80  W 

126.0304 

9 

NE  36£ 

19.20 

N  15.38 
E     5.74 

17.06W 
11.32  W 

262.3828 

10 

NE  22£ 

14.00 

N  12.93 
E     2.68 

8.64  W 
5.96  W 

111.7152 

11 

SE  76» 

12.00 

S     2.75 
E     5.84 

0.12  W 
5.72  E 

0.3300 

12 

SW  15 

10.85 

S    10.48 
W    1.40 

<.32  E 
2.92  E 

45.2736 

13 

SW  165 

10.12 

S     9.69 
W    1.46 

1.46  E 
0.00 

14.1474 

Contents  in  chains     -     -     - 

1285.1012 
178.0499 

178.0499 

1107.0513 

186 


COMPUTATION  OF  AREAS. 


The  foregoing  Field-look,  Method  II. 

It  it  needless  here  to  insert  the  columns  of  bearing  or  distances  in  chains,  they  being  the  same  as  before. 


No. 
St. 

Lat.  and 
halfDep. 

Merid. 
Dist. 

N.  Area. 

S.  Area. 

1 

N    3.54 
E    6.61 

6.61  E 
13.22  E 

23.3994 

2 

N    9.65 
E     1.80 

15.02  E 
16.82  E 

144.9430 

3 

0.00 
E    8.10 

24.92  E 
33.02  E 

4 

S  29.44 
W  9.74 

23.28  E 
13.54  E 

• 

685.3632 

5 

S     3.87 
W  7.76 

5.78  E 
1.98  W 

22.3686 

6 

N    9.00 
0.00 

1.98  W 
1.98  W 

17.8200 

7 

S     1.21 
W   5.77 

7.75  W 
13.52  W 

9.3775 

8 

N    6.94 
W  4.64 

18.16W 
22.80  W 

126.0303 

9 

N  15.38 
E    5.74 

17.06W 
11.32W 

262.3828 

10 

N  12.93 
E    2.68 

8.64  W 
5.96  W 

111.7152 

11 

S     2.75 

E    5.84 

0.12W 
5.72  E 

0.3300 

12 

S  10.48 
W   1.40 

4.32  E 
2.92  E 

45.2736 

13 

S     9.69 
W   1.46 

1.46  E 
0.00 

14.1474 

Area  in  chains,  ; 

178.0499 

1284.1012 
178.0499 

is  before, 

1107.0513 

•» 
COMPUTATION  OF  AREAS.  187 

A  Specimen    of  the    Pennsylvania   Method   of  CALCULA- 
TION; which  for  its  simplicity  and  ease  in  finding  the  Me-  . 
ridian  Distances  is  supposed  to  be  preferable  in  practice  to 
any  thing  heretofore  published  on  the  subject. 

Find,  in  the  first  place,  by  the  Traverse  Table,  the  lat.  and 
dep.  for  the  several  courses  and  distances,  as  already  taught ; 
and  if  the  survey  be  truly  taken,  the  sums  of  the  northings  and 
southings  will  be  equal,  and  also  those  of  the  eastings  and 
westings.  Then,  in  the  next  place,  find  the  meridian  distances, 
by  choosing  such  a  place  in  the  column  of  eastings  or  westings 
as  will  admit  of  a  continual  addition  of  one,  and  subtraction  of 
the  other ;  by  which  means  we  avoid  the  inconvenience  of 
changing  the  denomination  of  either  of  the  departures. 

The  learner  must  not  expect  that  in  real  practice  the  columns 
of  lat.  and  those  of  dep.  will  exactly  balance  when  they  are 
at  first  added  up,  for  little  inaccuracies  will  arise,  both  from  the 
observations  taken  in  the  field  and  in  chaining;  which  to  ad- 
just, previous  to  finding  the  meridian  distances,  we  may  observe, 
that  if  in  small  surveys  the  difference  amount  to  two-tenths 
of  a  perch  for  every  station,  there  must  have  been  some  error 
committed  in  the  field ;  and  the  best  way  in  this  case  will  be, 
to  rectify  it  on  the  ground  by  a  resurvey,  or  at  least  as  much 
as  will  discover  the  error.  But  when  the  differences  are  not 
within  those  limits,  the  columns  of  northing,  southing,  easting, 
and  westing  may  be  corrected  as  follows : 

Add  all  the  distances  into  one  sum,  and  say,  as  that  sum  is 
to  each  particular  distance,  so  is  the  difference  between  the  sums 
of  the  columns  of  northing  and  southing  to  the  correction  of 
northing  or  southing  belonging  to  that  distance :  the  corrections 
thus  found  are  respectively  additive  when  they  belong  to  the 
column  of  northing  or  southing  which  is  the  less  of  the  twor 
and  subtractive  when  they  belong  to  the  greater ;  if  the  course 
be  due  east  or  west,  the  correction  is  always  additive  to  the 
less  of  the  two  columns  of  northing  or  southing.  The  correc- 
tions of  easting  and  westing  are  found  exactly  in  the  same 
manner. 

The  following  example  will  sufficiently  illustrate  the  manne* 
of  applying  the  rule. 


% 


188 


COMPUTATION  OF  AREAS. 


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In  this  example,  the  sum  of  the  distances  is  791,  and  the  dif- 
ference between  the  columns  of  northing  and  southing  is  .4, 
also  the  first  distance  is  70  ;  say  then, 

791  :  70  :  :  .4  :  .04, 

which  fourth  proportional  .04  is  the  first  correction  belonging 
to  the  southing  53.6,  from  which  the  correction  *04  should  he 
subtracted. 


COMPUTATION  OF  AREAS.  189 

In  this  manner  the  several  corrections  of  the  southings 
53.6  )  {  .04 


29.1  \  are  found  to  be  ^  .09  )  respectively. 
135.7  i 

But  as  only  two  of  thes.e  corrections  amount  to  half  a  tenth,   ' 
we  must  use  .1  for  each  of  the  corrections  .09  and  .07,  and 
neglect  the  correction  .04  ;  thus  the  correct  southings  become 
53.6,  29.0,  135.6. 

In  like  manner  from  the  remaining  distances  we  obtain  to 

f   62.9)  ^.04 

the  northings  <    „  "    >  the  additive  corrections  <  ' 

(    00.0 )  (.07 

And  consequently,  by  neglecting  .04  and  .03,  and  using  .1 
for  each  of  the  two  .06  and  .07,  the  northings  when  corrected 
are  62.9,  101.2,54.0,  00.1. 

In  obtaining  these  corrections,  it  is  commonly  unnecessary 
to  use  all  the  significant  figures  of  the  distances  :  thus,  for  the 
ratio  of  791  to  70,  we  may  say,  as  80  to  7^ 

The  latitudes  and  departures  being  thus  balanced,  proceed 
to  insert  the  meridian  distances  by  the  above  method,  where 
we  still  make  use  of  the  same  field-notes,  only  changing  chains 
and  links  into  perches  and  tenths  of  a  perch.  Then  by  look- 
ing along  the  column  of  departure,  it  is  easy  to  observe,  that 
in  the  columns  of  eastings  opposite  station  9  all  the  eastings 
may  be  added,  and  the  westings  subtracted,  without  altering 
the  denomination  of  either.  Therefore,  by  placing  46.0,  the 
east  departure  belonging  to  this  station,  in  the  column  of  me- 
ridian distances,  and  proceeding  to  add  the  eastings  and  subtract 
the  westings,  according  to  the  rule  already  mentioned,  we  shall 
find  that  at  station  8  these  distances  will  end  in  0,  0,  or  a 
cipher,  if  the  additions  and  subtractions  be  rightly  made.  Then 
multiplying  the  upper  meridian  distance  of  each  station  by  its 
respective  northing  or  southing,  the  product  will  give  the  north, 
or  south  area,  as  in  the  examples  already  insisted  on,  and  which 
is  fully  exemplified  in  the  annexed  specimen.  When  these 
products  are  all  made  out  and  placed  in  their  respective  columns, 
their  difference  will  give  double  the  area  of  the  plot,  or  twice 
the  number  of  acres  contained  in  the  survey.  Divide  this 
remainder  by  2,  and  the  quotient  thence  arising  by  160  (the 
number  of  perches  in  an  acre),  then  will  this  last  quotient  ex- 
hibit the  number  of  acres  and  perches  contained  in  the  whole 
survey;  which  in  this  example  may  be  called  110  acres,  103 
perches,  or  110  acres,  2  roods,  23  perches. 


190 


COMPUTATION  OF  AREAS. 


FIELD-NOTES  of  the  two  foregoing  methods,  as  practised 
in  Pennsylvania. 

Cast  up  by  perches  and  tenths  of  a  perch. 


N. 

Courses. 

Dist. 

N. 

S. 

E. 

W. 

M.  D- 

N.  Area. 

S.  Area. 

1 

N  750  00'  E 

54.8 

14.2 

52.9 

235.3 

288.2 

3341.26 



2 
3 

N  20.30  E 

41.2 

38.6 

14.4 

302.6 
317.0 

11680.36 

East. 

64.8 

64.8 

381.8 
446.6 

4 
5 
6 

S  33.30  W 

141.2 

117.7 

77.9 

368.7 
290.8 

43395.99 

S  76.00  W 

64.0 

15.5 

62.1 

228.7 
166.6 

3544.85 

North. 

36.0 

36.0 

166.6 
166.6 

5977.60 

7 
8 
9 
10 

S  84.00  W 

46.4 

4.9 

46.1 

120.5 
74.4 

590.45 

N  53.15  W 

46.4 

27.8 

37.2 

37.2 
00.0 

1034.16 

N  36.45  E 

76.8 

61.5 

46.0 

46.0 
92.0 

2829.00 

N  22.30  E 

56.0 

51.7 

21.4 

113.4 
134.8 

5862.78 

11 
12 

S  76.45  E 

48.0 

11.0 

46.7 

181.5 

228.2 

1996.50 

S  15.00  W 

43.4 

41.9 

11.2 

217.0 
205.8 

9092.30 

13 

S  16.45  W 

40.5 

38.8 

11.7 

194.1 

182.4 

7531.08 

f 

229.8 

229.8 

246.2 

246.2 

30745.16 

66151.17 
30745.16 

Area  i 

2 

35406.01 

i  perches  177030.05 

COMPUTATION  OF  AREAS. 


101 


Note. — Tn  the  foregoing  methods  the  first  meridian  passes 
through  the  map ;  but  as  it  is  more  convenient  to  have  it  pass 
through  the  extreme  east  or  west  point  of  the  same,  I  have 
given  the  folio  wing -example  to  illustrate  this  method. 

Of  computing  the  area  of  a  survey  by  having  the  bearings  and  distances 
given,  geometrically  considered  and  demonstrated. 

Let  BCDEFGHA,  pi.  14,  fig.  1 1,  represent  the  boundary  of 
a  survey  of  which  the  following  field-notes  are  given ;  it  is  re- 
quired to  find  the  area. 


EXAMPLE. 


* 

Sides  of 
the  land. 

Bearings. 

Length"  in 
chains. 

BC 

East. 

4.00 

CD 

N9°  E 

4.00 

DE 

S69  E 

5.56 

EF 

S  36  E 

7.00 

FG 

S42W 

4.00 

GH 

S75W 

10.00 

HA 

N39W 

7.50 

AB 

N42E 

5.00 

RULE    I. 

Find  the  difference  of  latitude  and  departure  answering  .to 
each  course  and  distance  by  the  Traverse  Table  or  right- 
angled  plane  trigonometry,  according  to  the  directions  already 
given,  and  place  them  under  the  succeeding  columns  North  or 
South,  East  or  West,  according  as  they  are  north  or  south, 
east  or  west ;  then  if  the  survey  does  not  close,  correct  the 
errors  fey  saying,*  as  the  sum  of  all  the  distances  is  to  each 

*  This  arithmetical  rule  was  given  by  Mr.  Bowditch  in  his  solution  of 
Mr.  Patterson's  question  of  correcting  a  survey  in  No.  4  of  the  Analyst. 
Also,  the  editor,  Dr.  Ad'.ain,  has  given  precisely  the  same  practical  rule, 


192  COMPUTATION  OF  AREAS. 

particular  distance,  so  is  the  whole  error  in  departure  to  the 
correction  of  the  corresponding  departure,  each  correction  be- 
ing so  applied  as  to  diminish  the  whole  error  in  departure  :  pro- 

in  his  elegant  solution  of  the  said  question,  analytically  demonstrated.  As 
the  demonstration  of  this  important  rule  may  give  great  satisfaction  to 
those  who  have  not  an  opportunity  of  seeing  the  Analyst,  I  have  inserted 
Mr.  Bowditch's  demonstration  of  said  rule,  which  is  as  follows,  viz. 

Demonstration  1.  That  the  error  ought  to  be  apportioned  among  all 
the  bearings  and  distances. 

2.  That  in  those  lines  in  which  an  alteration  of  the  measured  distance 
would  tend  considerably  to  correct  the  error  of  the  survey,  a  correction 
ought  to  be  made  ;  but  when  such  an  alteration  would  not  have  that  ten- 
dency, the  length  of  the  line  ought  to  remain  unaltered. 

3.  In  the  same  manner,  an  alteration  ought  to  be  made  in  the  observed 
bearings,  if  it  would  tend  considerably  to  correct  the  error  of  the  survey, 
otherwise  not. 

4.  In  cases  where  alterations  in  the  bearings  and  distances  will  both  tend 
to  correct  the  error  it  will  be  proper  to  alter  them  both,  making  greater  or 
less  alterations  according  to  the  greater  or  less  efficacy  in  correcting  the 
error  of  the  survey. 

5.  The  alterations  made  in  the  observed  bearing  and  length  of  any  one 
of  the  boundary  lines  ought  to  be  such  that  the  combined  effect  of  such 
alterations  may  tend  wholly  to  correct  the  error  of  the  survey. 

Suppose  now  that  ABODE  (pi.  14,  fig.  12)  represent  the  boundary  lines 
of  a  field,  as  plotted  from  the  observed  bearings  and  lengths,  and  that  the  last 
point  E,  instead  of  falling  on  the  first  A,  is  distant  from  it  by  the  length  AE. 
TJhe  question  will  therTbe,  what  alterations  BB',  CC",  DD'",  &c.  must 
be  made  in  the  positions  of  the  points  JB,  C,  D,  &c.  so  as  to  obtain  the 
most  probable  boundaries  AB'C"D'"A1  If  AB'  be  supposed  to  be  the 
most  probable  bearing  and  length  of  the  first  boundary  line,  the  point  B 
would  be  moved  through  the  line  BB\  and  the  following  points  C,  Z),  E 
would  in  consequence  thereof  be  moved  in  equal  and  parallel  directions  to 
C',  D',  E',  and  the  boundary  would  become  AB'C'D'E'.  Again,  if  by 
correcting  in  the  most  probable  manner  the  error  in  the  observed  bearing 
and  length  of  BC  (or  -B'C'),  the  point  C'  be  moved  to  C",  the  points  I)' 
and  E'  would  be  moved  in  equal  and  parallel  directions  to  D"  and  E",  and 
the  boundary  line  would  become  AB'C"D"E".  In  a  similar  manner,  if 
by  correcting  the  probable  error  in  the  bearing  and  length  of  CD  (or  C"D'") 
the  point  D"  be  moved  to  D'",  the  point  E"  would  be  moved  in  an  equal 
and  parallel  direction  to  E'",  and  the  boundary  would  become  A  B'  C"D"'E"". 
Lastly,  by  correcting  the  probable  error  in  the  bearing  and  length  of  the 
line.DE  (or  D"'E'")  the  true  boundary  AB'C"D'"A  would  be  obtained. 
If  we  suppose  the  lines  BB'CC"DD'",  &c.  to  be  parallel  toAE,  it  would 
satisfy  the  second,  third,  fourth,  and  fifth  of  the  preceding  principles.  For 
the  change  of  position  of  the  points  B,  C,  &c.  being  in  directions  parallel 
to  AE,  the  whole  tendency  of  such  change  would  be  to  move  the  point  E 
directly  towards  A,  conformably  to  the  fifth  principle  ;  and  by  inspecting 
the  figure,  it  will  appear  that  the  second,  third,  and  fourth  principles  would 
also  be  satisfied.  For,  in  the  first  place,  it  appears  that  the  bearing  of  the 
first  line  AB  would  be  altered  considerably,  but  the  length  but'  little.  This 
is  agreeable  to  those  principles,  because  an  increase  of  the  distance  AB 
would  move  the  point  E  in  the  direction  Eb  parallel  to  AB,  and  an  altera- 
tion in  the  bearing  would  move  it  in  the  direction  Eb'  perpendicular  to 


COMPUTATION  OF  AREAS.  193 

ceed  the  same  way  for  the  corrections  in  latitudes.  These 
corrections  being  applied  to  their  corresponding  differences  of 
latitude  and  departure,  that  is,  add  when  of  the  same  name  and 

AB.  Now  the  former  change  would  not  tend  effectually  to  decrease  the 
distance  AE,  but  the  latter  would  be  almost  wholly  exerted  in  producing 
that  effect.  Again,  the  length  of  the  line  BC  would  be  considerably 
changed-  without  altering  essentially  the  bearing ;  the  former  alteration 
would  tend  greatly  to  decrease  the  distance  AE,  but  the  latter  would  not 
1  produce  so  sensible  an  effec,t.  Similar  remarks  may  be  made  on  the 
changes  in  the  other  bearings  and  distances,  but  it  does  not  appear  to  be 
necessary  to  enter  more  largely  on  this  subject. 

It  remains  now  to  determine  the  proportion  of  the  lines  BB',  CC", 
DD'",  &c.  To  do  this  we  shall  observe,  that  in  measuring  the  lengths 
of  any  lines  the  errors  would  probably  be  in  proportion  to  their  lengths. 
These  supposed  errors  must,  however,  be  decreased  on  those  lines  where 
the  effect  in  correcting  the  error  of  the  survey  would  be  small,  by  the 
second  and  fourth  principles. 

In  observing  the  bearings  of  all  the  boundary  lines  equal  errors  are 
liable  to  be  committed  ;  however,  it  will  be  proper,  by  the  third  and  fourth 
principles  to  suppose  the  error  greater  or  less  in  proportion  to  the  greater 
or  less  effect  it  would  produce  in  correcting  the  error  of  the  survey. 

Now  the  error  of  an  observed  bearing  being  given,  as  for  example  GFI 
(pi.  14,  fig.  13),  the  change  of  position  GI  of  the  end  of  the  line  G  would 
be  proportional  to  the  length  of  the  line  FG  (=FI),  so  that  the  supposed 
errors  both  in  the  length  and  in  the  bearing  of  any  boundary  line  would 
produce  changes  in  the  position  of  the  end  of  it  proportional  to  its  length. 
There  appears,  therefore,  a  considerable  degree  of  probability  in  supposing 
the  lines  BB',  C'C",  D'D'",  &c.  to  be  respectively  proportional  to  the 
lengths  of  the  boundary  lines  AB,  BC,  CD,  &c.  The  main  point  to  be 
ascertained  before  adopting  this  hypothesis  is,  whether  a  due  proportion 
of  the  error  of  the  survey  is  thrown  on  the  bearings  and  lengths  of  the 
sides.  Now  it  is  plain  by  this  hypothesis  that  the  error  in  any  boundary 
line  is  supposed  to  be  wholly  in  the  bearing  if  the  line  be  perpendicular 
to  AE,  and  wholly  in  its  length  when  parallel  to  AE  ;  and  if  the  length 
be  the  same  in  both  cases,  the  change  of  position  of  the  end  of  the  line 
would  in  both  cases  be  exactly  equal.  Thus,  if  FGH  be  the  boundary 
line,  GI  the  change  of  position  of  the  point  B  in  the  former  case,  and  OH 
in  the  latter,  we  should  in  this  hypothesis  have  GI=GH. 

To  show  the  probability  of  this  hypothesis  it  may  be  observed,  that  in 
measuring  the  lengths  of  a  line  FGH  of  six  or  eight  chains  of  fifty  links 
each,  an  error  of  one  link  might  easily  be  committed  by  the  stretching  of 
the  chain  or  the  unevenness  of  the  surface.  This  error  would  be  about 
~j  of  the  whole  length.  If  we,  therefore,  suppose  GI  to  be  ^—  of  FG, 
the  angle  GFI  would  be  about  10'.  Now,  with  such  instruments  as  are 
generally  made  use  of  by  surveyors,  it  is  about  as  probable  that  an  error 
of  KX  was  made  in  the  bearing  as  that  the  above  error,  — |—  part,  was  made 
in  measuring  the  length.  We  shall  therefore  adopt  it  as  a  principle,  that 
the  most  probable  way  of  apportioning  the  error  of  the  survey  AE  is  to 
take  BB1,  C1  C",  ,D"D"',  &c.  respectively  proportional  to  the  boundary 
lines  AB,  BC,  CD,  &c. 

Hence  the  following  practical  rule  for  correcting  a  survey  geometrically. 
Draw  the  boundary  lines  ABCDE  by  means  of  the  observed  bearings  and 


194 


COMPUTATION  OF  AREAS. 


subtract  when  of  different  names,  then  the  corrected  difference 
of  latitude  and  departure  will  be  obtained,  and  the  table  will 
stand  thus : 


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lengths,  and  find  the  error  of  the  survey  AE,  and  let  the  quotient  of  AE 
divided  by  the  sum  of  all  the  lines  AB,  BC,  CD,  DE  be  represented  by 
r ;  through  the  angular  points  &,  C,  .D,  &c.  draw  the  fines  BB',  CC", 
&c.  parallel  to  AE,  and  in  the  same  direction  that  A  bears  from  E.  Take 
BB'=rxAB,  CC"=rX  (AB+BC),  DD'"=rX  (AB+BC+CD),  &c. 


COMPUTATION  OF  AREAS.  195 


The  errors  being  corrected  thus 


''  ''  '      :  *The  corrections  of  difference  of  lat. 


«    :  *no^ 
47:4::.28:.02j      as  in  Tabie  L 

.  &c.  ) 


&c, 

'  *      *  '      )  The  corrections  of    departure  as  in 
As  47:4::   22:   02  £      Table  j 

&C.    &C.  } 

The  latitudes  and  departures  being  thus  balanced,  it  is  neces- 
sary to  calculate  the  several  meridian  distances,  in  order  to  com- 
pute the  area  of  the  survey. 

As  beginning  at  the  most  easterly  or  most  westerly  point  of 
the  survey  admits  of  a  continual  addition  of  the  one  and  sub- 
traction of  the  other,  the  most  easterly  or  most  westerly  point 
can  be  easily  discovered  from  the  foregoing  table,  thus  : 

The  first  departure  corrected  is  3.98,  which  is  the  meridian 
distance  of  the  second  point  of  the  survey  from  the  first,  to 
which  add  0.61  the  next  dep.  corrected,  and  their  sum  is  4.59, 
the  meridian  distance  of  the  third  points  of  the  survey  from  the 
first;  andinlike  manner  4.59+5.17=9.76=  the  meridian  dis- 
tance of  the  fourth  point  from  the  first,  and  9.76-f-4. 08  =  13.84 
=  the  meridian  distance  east  of  the  fifth  point  from  the  first ; 
after  the  same  manner,  continue  to  add  the  dep.  when  east, 
but  subtract  when  west :  the  next  dep.  is  west,  therefore  13.84 
— 2.70  =  11.14  =  the  meridian  distance  of  the  sixth  point  from 
the  first,  and  11.14  —  9.71  =.43=  the  next.  Now  the  next  de- 
parture is  4.76,  which  is  west,  and  1.43  is  the  meridian  distance 
of  the  seventh  point  from  the  first,  which  is  east ;  therefore 
4.76  —  1.43=3.33=  the  meridian  distance  of  the  eighth  point* 
from  the  first ;  as  3.33  is  the  greatest  meridian  distance  west  of 
the  eighth  point  of  the  survey  from  the  first,  because  the  next 
departure  is  east  3.33;  then,  3.33 — 3.33=0,  which  closes  tlje 
survey :  consequently,  the  eighth  point  of  the  survey  is  the  most 
westerly  point,  and  for  the  same  reason  as  13.84  is  the  greatest 
meridian  distance  east,  which  is  the  meridian  distance  of  the 
fifth  point  of  the  survey.  In  like  manner,  the  most  easterly  or 

Then  through  the  points  A,  B\  C",  D'",  &c.  draw  the  corrected  boundary 
lines  ABC  DA,  which  being  determined,  the  area  may  be  found  by  dividing 
the  figure  into  triangles  in  the  usual  method. 

The  proportional  parts  BB',  CC",  &c.  may  be  found  expeditiously  by 
means  of  a  table  of  difference  of  latitude  and  departure,  by  finding  the 
page  where  the  sum  of  the  lines  AB+BC-\-CD-\-DE  in  the  distance 
column  corresponds  to  AE  in  the  departure  or  difference  of  latitude 
column  ;  then  find  AB,  AB-\-BC,  &c.  in  the  distance  column,  and  the 
corresponding  numbers  will  be  equal  to  BB',  CC",  DD"',  &«.  respect- 
ively. 

I  2 


196  COMPUTATION  OF  AREAS. 

most  westerly  point  of  the  survey  can  be  found  by  beginning  at 
any  other  point. 

After  the  most  easterly  or  most  westerly  point  of  the  survey 
is  discovered,  call  that  point  the  first  station,  and  proceed  to 
find  the  meridian  distances  for  the  several  lines  in  the  order  in 
which  they  were  surveyed;  that  is,  the  first  dep.  will  be  the 
first  meridian  distance,  which  place  in  the  column  of  meridian 
distances  opposite  the  said  departure  ;  to  the  same  meridian  dis- 
tance add  the  said  departure,  to  which  sum  add  the  next  de- 
parture if  it  be  of  the  same  name  with  the  foregoing  departure, 
but  subtract  if  it  be  of  a  different  name,  which  sum  or  differ- 
ence call  the  next  meridian  distance,  and  set  it  in  the  column 
of  meridian  distances  opposite  the  departure  last  used ;  and  in 
like  manner,  continue  to  add  the  departure  twice  when  of  the 
same  name,  but  if  of  a  different  name  subtract  twice,  and  the 
last  meridian  distance  will  be  zero,  if  the  additions  and  sub- 
tractions are  rightly  performed ;  because  the  sum  of  the  north- 
ings is  equal  to  the  sum  of  the  southings  after  the  survey  is 
corrected,  which  is  evident  from  theo.  1,  and  the  foregoing 
table.  Then,*  multiplying  the  upper  meridian  distance  of  each 
station  by  the  corresponding  northing  or  southing,  and  place  the 
product  in  the  north  or  south  area,  according  as  the  latitude  is 
north  or  south,  the  difference  of  the  sum  of  these  products 
will  give  twice  the  area,  half  of  which  gives  the  area  of  the 
survey. 

The  most  westerly  point  of  the  survey  being  made  the  first 
station,  and  the  several  meridian  distances  being  calculated,  &c., 
the  foregoing  table  will  stand  thus : 

*  Demonstration.  Let  N&  be  a  meridian  passing  through  the  most 
westerly  station  from  the  points  B,  C,  D,  E,  F,  G,  and  H ;  let  fall  the 
perpendiculars  Bb,  Cb,  Dd,  EC,  Fe,  Gf,  and  HI,  on  the  meridian  NS. 

Now,  if  from  the  area  of  the  figure  dDEFGHI  the  area  of  the  figure 
dDCBAHI  be  taken,  there  remains  the  area  of  the  survey:  The  area  of 
the  multangular  figure  dDEFGHI  is  equal  to  the  sum  of  the  areas  of  the 
trapezoids  of  which  it  is  composed,  viz.  dDEc,  cEFe,  eFGf,  audfGHI; 
but  (by  prob.  10),  (AD-\-cE]  X  dc=  twice  the  area  of  the  trapezoid  dDEc  ; 
and  dD-\-cE  equal  to  the  sum  of  the  meridian  distances  of  the  points  D  and 
E  from  the  first  meridian  line  NS,  and  dc  or  dg  =  the  southing  of  the 
point  E  from  the  point  D.  In  like  manner  the  area  of  every  other  trape- 
zoid is  found  :  but  these  are  the  south  column  areas  :  thab  is,  (dD-\-c E)  X 
dc+(cE+eF)Xce-\-(eF-\-fG)  X  ef-{-(f  G-\-IH)  Xfl=-  twi|e  the  area  of 
the  figure  dDEFGHI  =  the  sum  of  the  south  area  column.  And',  in  like 
manner,  we  demonstrate  that  (dD-\- bC)  X  db-^-bB  X  bA+A^X  IH=  twice 
the  area  of  the  figure  dDCBAHI  =^  the  north  area  column ;  therefore, 
(dD+cE)  X  dc+(cE+eF)  X  ce+  (eF+fG)  X  ef+  (fG+  lit)  X/I  —  [(dD 
•}-bC)Xdb+bBxbA-{-AIxIH]  =  twice  the  area  of  the  survey ;  conse- 
quently, the  sum  of  the  south  area  column  —  the  sum  of  the  north  area 
column  =  twice  the  area  of  the  survey.  Q.  E.  D. 


•! 


COMPUTATION  OF  AREAS 


<«  I  ss  I  RE  !  fc; 

5  Tj<  -*  00  O>  O>        |      Tjt  I 


rl         I        CO 


«         '       «         I       O 


^  I     8    I    8    |    §    |    8.    |    S    |    3    |    8    [    8 


§        g 


*  ,  I    d         « 


s      a 


RULE    H.J 

The  difference  of  latitude  and  departure  being  found  and  cor- 
rected as  in  the  preceding  rule. 

*  This  is  not  the  first  station  in  t^ie  actual  survey,  hut  only  the  most 
westerly  point  of  the  survey  as  calculated  by  the  foregoing  method  from 
the  field-notes,  which,  for  convenience'  sake,  I  call  the  first  station  in  mak- 
ing out  this  table. 

t  The  meridian  distances  in  this  column  are  the  sum  of  two  adjacent 
meridian  distances  ;  but  at  the  most  westerly  point  the  meridian  distance 
is  nothing,  hence  the  first  dep.  is  the  first  meridian  distance,  and,  in  like 
manner,  the  last  dep.  is  the  last  meridian  distance. 

J  Demonstration.  Let  us  consider  that  every  tract  of  land  has  an  ex- 
treme southerly  point,  as  H ;  and  we  reckon  so  much  as  any  other  point  is 
distant  from  the  east  and  west  line  IK  (PL  14,  fig.  11),  that  passes  through 


108  COMPUTATION  OF  AREAS. 

As  beginning  at  the  most  northerly  or  most  southerly  point 
of  the  survey  admits  of  a  continual  addition  of  the  one  and 
subtraction  of  the  other,  make  choice  of  either  of  these  points 
in  order  to  calculate  the  area  of  the  survey. 

1.  It  is  necessary  to  calculate  the  several  latitudes  in  order 
to  find  the  most  northerly  or  most  southerly  point  of  the  survey, 
which  may  be  done  from  Table  I.,  thus : 

The  first  lat.  is  .02  south,  which  is  the  difference  of  latitude 
between  the  second  point  of  the  survey  and  the  first,  when  the 
survey  is  corrected  from  the  next  departure  3.93,  which  is  N., 
subtract  .02  and  their  difference  3.91  is  equal  to  the  difference 
of  latitude  between  the  third  point  and  the  first,  which  is  N.,  and 
3.91 — 2.02  —  1.89  =  the  difference  of  lat.  between  the  fourth 
point  and  the  first ;  which  is  also  N.  .But  as  the  next  differ- 
ence of,  lat.  is  south,  therefore  5.71 — 1.89=3.32  =  the  differ- 
ence of  lat.  S.  between  the  fifth  point  and  ihe  first ;  and  3.82+ 
2.99=6.81=  the  difference  of  lat.  S.  between  the  sixth  point 
and  the  first ;  and  6.81+2.65=9.46  =  the  difference  of  lat.  S. 
between  the  seventh  point  and  the  first ;  and  9.46 — 5.77  =  3.69 
=  the  difference  of  lat.  S.  between  the  eighth  point  and  the 
first;  and  3.69 — 3.69=0  ;  hence  it  is  evident  that  9.46  is  the 
greatest  lat.  S.  =  the  difference  of  lat.  between  the  seventh 
point  and  the  first;  therefore,  the  seventh  point  of  the  survey 
is  the  most  southerly  point ;  and,  in  like  manner,  3.91  =  the 
difference  of  lat.  between  the  third  point  and  the  first,  is  the 
greatest  lat.  N.  ;  hence,  the  third  point  is  the  most  northerly 
point  of  the  survey. 

H,  to  be  its  latitude  north,  or  the  difference  of  latitude  between  the  points 
H  and  A  ;  EL  the  lat.  of  B ;  CM  the  lat  of  C  ;  &c. 

Thus,  if  from  the  contents  of  the  figure  MB  CDEFK,  the  contents  of  the 
figure  FKIAHG  be  subtracted,  the  remainder  will  be  the  area  of  the  survey. 

The  multangular  figure  IABCDEFK  is  composed  of  all  these  trape- 
zoids,  viz.  IABL,  B  CML,  CDNM,  EOND,  and  FKOE ;  but  (by  Prob.  10) 
(1A  -\-LB)  X  ZL  =  twice  the  area  of  the  trapezoid  IABL,  and  (LB-\-CM) 
X  LM  =  twice  the  area  of  the  trapezoid  B  CML,  and  so  for  the  rest ;  and 
IA-\-LB  =  the  sum  of  the  northings  of  the  points  A  and  B  from  the  line 
IK,  and  IL  =  the  easting  of  the  pqkit  B  from  the  point  A.  In  like  man- 
ner the  area  of  every  other  trapezom  is  found  ;  but  these  are  the  east  col- 
umn areas,  that  is,  (IA-\-BD X IL  +(#!+  CM) X LM-}-(CM+DN) X 
MN-)-(DN+EO)xNO-}-(EO-\-FK)  X  OK=  twice  the  area  of  the  figure 
IAB  CDEFK  =  the  sum  of  the  east  area  column.  And  in  like  manner  we 
demonstrate  that  (FK-\-PG)xPK  =  twice  the  area  of  the  trapezoid 
FKPG ;  but  FK+PG  =  the  lat.  of  F+  the  lat,  of  G  and  PK  =  the  dep.  or 
westing  of  the  point  G  from  the  point  F,  and  PGxPH=  twice  the  area 
of  the  triangle  PGH,  and  lAxIH  —  twice  the  area  of  the  triangle  IAH\ 
hence  (FK+PG)xPK+PGxPH+lAxIH  =  twice  the  area  of  the 
figure  FKIAGH  =  the  sum  of  the  west  area  column.  Therefore  (/.#+ 
JgL)  X IL+  (BL-4-CM)  x  LM+  ( CM+DN)  x  MN+  (DN+EO)  x  NQ 
+(EO+FK)  x  OK-  ((FK+PG)  X  PK+PG  X  PH+IA  X IH]  =  twice 
the  area  of  the  survey  ;  consequently,  the  sum  of  the  east  area  column  — . 
the  sum  of  the  west  area  column  —  twice  the  area  of  the  survey.  Q.  E.  IX 


COMPUTATION  OF  AREAS. 


199 


Now,  by  calling  the  most  southerly  point  of  the  survey  the 
first  station,  and  proceeding  to  find  the  latitudes  for  the  several 
lines  in  the  order  in  which  they  were  surveyed ;  that  is,  the  first 
difference  of  lat.  will  be  the  first  lat.,  which  place  in  the  column 
of  latitudes,  opposite  the  said  difference  of  latitude ;  to  the 
same  lat.  add  the  said  difference  of  lat.,  to  which  sum  add 
the  next  difference  of  lat.  if  it  be  of  the  same  name,  but  sub- 
tract if  of  a  different  name,  and  place  it  in  the  column  of  lati- 
tudes ;  in  like  manner  continue  to  add  or  subtract  the  difference 
of  lat.  twice,  and  the  last  lat.  comes  out  nothing,  if  the  addi- 
tions and  subtractions  are  rightly  performed.  Multiply  each 
of  the  upper  numbers  in  the  column  of  latitudes  by  the  corres- 
ponding dep.,  and  place  the  products  in  the  column  of  east  or 
west  area,  according  as  the  dep.  is  E.  or  W.  The  difference 
of  these  columns  will  be  equal  to  twice  the  area,  half  of  which 
will  give  the  area  of  the  survey ;  as  in  the  following  table. 


Il   i 


!  §»  I  §  oq     53  ?• 
IS  I  22      gi§ 


MS 


3        3 


S    I   § 


"*        2 


200  OF  OFFSETS. 

Each  of  the  numbers  in  the  column  of  latitudes  is  twice  the 
mean  latitude  of  two  adjacent  latitudes  ;  but  at  the  most  south- 
erly point  the  latitude  is  nothing ;  hence  the  first  difference  of 
latitude  is  the  first  lat.,  and  in  like  manner  the  last  difference 
of  lat.  is  the  last  latitude.  It  is  also  to  be  remarked  that  the 
first  station  used  in  this  table  is  not  the  first  station  in  the 
actual  survey,  but  the  most  southerly  point  of  the  survey,  as 
calculated  by  the  foregoing  method  from  Table  I. 


SECTION  IV. 
OF  OFFSETS. 

IN  taking  surveys  it  is  unnecessary  and  unusual  to  make  a 
station  at  every  angular  point,  because  the  field-work  can  be 
taken  with  much  greater  expedition  by  using  offsets  and  in- 
tersections, and  with  equal  certainty  ;  especially  where  creeks, 
&c.  bound  the  survey. 

Offsets  are  perpendicular  lines  drawn  or  measured  from  the 
angular  points  of  the  land,  that  lie  on  the  right  or  left-hand  to 
the  stationary  distance,  thus  : 


Let  the  black  lines  represent  the  boundaries  of  a  farm  or 
township  ;  and  let  1  be  the  first  station  :  then  if  you  have 
a  good  view  to  2,  omit  the  angular  points  between  1  and  2,  and 
take  the  bearing  and  length  of  the  stationary  line  1,  2,  and  in- 
sert them  in  your  field-book  ;  but  in  chaining  from  1  to  2,  stop 
at  d  opposite  the  angular  point  «,  and  in  your  field-book  insert 
the  distance  from  1  to  d,  which  admit  to  be  4ch.  25/.,  as  well 
as  the  measure  of  the  offset  ad,  which  admit  to  be  Ich. 
12/.,  thus  :  by  the  side  of  your  field-book,  in  a  line  with  the 
first  station,  Say  at  4ch.  25/.  L.  Ich.  12/.,  that  is,  at  4ch.  251. 
there  is  an  offset  to  the  left-hand  of  Ich.  121. 

This  done,  proceed  on  your  distance  line  to  e  opposite  to  the 
angle  &,  and  measure  eb  ;  supposing  then  le  to  be  7ch.  40/., 
and  eb  3ch.  401.,  say  (still  in  a  line  with  the  first  station  in 
your  field-book)  at  7ch.  401  L.  3ch.  401,  that  is,  at  7ch. 
401  there  is  an  offset  to  the  left  of  3cA.  40/.  ;  proceed  then 
with  your  distance  line  to/  opposite  to  the  angle  c,  and  measure 
fc'j  suppose  then  If  to  be  I3ch.  and/c  Ich.  25/.,  say,  in  the 
same  line  as  before,  at  13cA.  L.  leh.  25L  Then  proceed  from 
f  to  2,  and  you  will  have  the  measure  of  the  entire  stationary 
line  1,  2,  which  insert  in  its  proper  column  by  the  bearing. 


OF  OFFSETS.  201 

In  taking  offsets,  it  is  necessary  to  have  a  perch  chain,  or 
a  staff  of  half  a  perch,  divided  into  links  for  measuring  them ; 
for  by  tliis  means  the  chain  in  the  stationary  line  is  undis- 
turbed, and  the  number  of  chains  and  links  in  that  line  from 
whence,  or  to  which,  the  offsets  are  taken,  may  be  readily 
known. 

Having  arrived  at  the  second  station,  if  you  find  your  view 
will  carry  you  to  3,  take  the  bearing  from  2  to  3,  and  in  mea- 
suring the  distance  line,  stop  at  I  opposite  g ;  admit  21  to  be 
4ch.  10/.,  and  the  offset  Ig  Ich.  20Z.,  then  in  a  line  with  the 
second  station  in  your  fiefd-book,  say  at  4ch.  10/.  R.  Ich.  20/., 
that  is,  the  offset  is  a  right-hand  one  of  Ich.  201.  Again,  at 
fn,  which  suppose  to  be  Wch.  251.  from  2,  take  the  offset  mh 
of  Ich.  15/.,and  in  a  line  with  the  second  station,  say  at  lOeA. 
251.  R,  Ich.  151.  -In  the  same  line,  when  you  come  to  the 
boundary  at  t,  insert  the  distance  2z,  13cA.  10/.,  thus,  at  13cA. 
WL  0 ;  that  is,  at  13cA.  10/.  there  is  no  offset.  At  n,  which 
is  I5ch.  from  2,  take  the  offset  nk  45/.,  and  still  opposite  to 
the  second  station  say  at  I5ch.  L.  451. 

Let  the  line  3,  6  represent  the  boundary  which  by  means 
of  water,  briers,  or  any  other  impediment,  cannot  be  measured. 
In  this  case  make  one  or  more  stations  within  or  without  the 
land,  where  the  distances  may  be  measured,  and  draw  a  line 
from  the  beginning  of  the  first  to  the  end  of  the  last  distance, 
thus  :  make  stations  at  3,  4,  and  5,  take  the  bearings,  and  mea- 
suring the  distances  as  usual,  which  insert  in  your  field-book, 
and  draw  a  mark  like  one  side  of  a  parenthesis,  from  the  third 
to  the  fifth  station,  to  show  that  a  line  drawn  from  the  third 
station  to  the  farthest  end  of  the  fifth  stationary  line  will  ex- 
press the  boundary.  Thus, 

No.  Sta.  Deg.  ch.  L 

[3  172^  5.45 

4  200  13.25 

5  250  3.36 

Suppose  the  point  p  of  the  boundary  to  be  inaccessible  by 
means  of  the  lines  6p  or  p7  being  overflowed,  or  that  a  quarry, 
furze,  &c.  might  prevent  your  taking  their  lengths :  in  this  case 
take  the  bearing  of  the  line  6,  7,  which  insert  opposite  to  the 
sixth  station  in  your  field-book  with  the  other  bearing ;  then 
direct  the  index  to  the  point  p1  and  insert  its  bearings  on  the 
left  side  of  the  field-book,  opposite  to  the  sixth  station,  annexing 
thereto  the  words  Int.  for  boundary ;  and  having  measured  and 
inserted  the  distance  6,  7,  set  the  index  in  the  direction  of  the 
line  7p,  and  insert  its  bearing  on  the  left  of  the  seventh  station 
13 


202 


OF  OFFSETS. 


of  the  field-book,  annexing  thereto  the  words  ipt.  for  boundary  ? 
the  crossing  or  inter  section  of  these  two  bearings  will  deter- 
mine the  point  p,  and  of  course  the  boundary  6j»7  is  also  de- 
termined. 

If  your  view  will  then  reach  in  the  first  station,  take  its  bear- 
ing, stationary  line,  and  offsets  as  before,  and  you  have  the 
field-book  completed.  Thus, 

The  Field-book. 


Remarks  and  Inter. 

N. 
St. 

Deg. 

ch.  I. 

OFFSETS. 

318  Int.  to  a  tower 

1 

358 

22.12 

At  4ch.  25LL.lch.  121 

at-7cA.40Z.L.3cA.40/. 

at  13cA.  L.  Ich.  251 

231|Int.  to  ditto 

2 

297f 

22.12 

At4cA.  10/.R.  lcA.20Z. 

at  lOcA.  251   R.  IcA. 

511.   at   13cA.  IQL  0. 

at  \5ch.  L.  45Z. 

r 

3 

1721 

5.45 

4 

200 

13.25 

I 

5 

250 

3.36 

1551  Int.  for  bound. 

6 

125 

15.15 

Ktlch.2QLL.2ch.20L 

274  Int.  for  ditto. 

7 

1051 

15.10 

at7cA.45/.L.2cA.32Z. 

atllcA.25Z.O.atl2cA. 

251.  R.  36Z. 

Close  at  the  first  station. 

If  you  would  lay  down  a  tower,  house,  or  any  other  remark- 
able object  in  its  proper  place,  from  any  two  stations  take 
bearings  to  the  object,  and  their  intersection  will  determine  the 
place  whe^p  you  are  to  insert  it,  in  the  manner  that  the  tower 
is  set  out  in  the  figure,  from  the  intersection  taken  at  the  first 
and  second  stations  of  the  above  field-book. 

A  protraction  of  this  will  render  all  plain,  on  which  lay  off 
all  your  offsets  and  intersections,  and  proceed  to  find  the  con- 
tents by  any  of  the  methods  in  section  the  fourth. 


OF  OFFSETS.  203 

.    The'foregoing  Field-book  may  be  otherwise  kept,  thus : 


Remarks  and  Intersection. 

No. 
St. 

Deg. 

L.  hand 
Offset. 
ch.L 

Dist. 

ch.l 

R.hand 
Offset. 
ch.  I 

318  Int.  to  a  tower    - 

1 

358 

1.12 

4.25 

3.40 

7.40 

1.25 

13.00 

• 

232^  Int.  for  ditto  -    - 

22.12 

2 

297£ 

4.10 
10.25 

1.20 
1.15 

13.10 

•'•'  •             '  **-*     *;'  •  j 

0.45 

15.00 

21.21 

3 

4 

1721 
200 

5.45 
13.25 

5 

250 

** 

3.36 

155^  Int.  for  boundary 
274  Int.  for  boundary 

6 

125 

15.15 

7 

105 

2.20 
2.32 

1.20 
7.45 

11.25 

12.25 

0.36 

15.10 

, 

How  to  cast  up  offsets  by  the  pen. 

PL.  11.  Jig.  2. 

1,  2— l/=2/,2/— -le=fe,  le—ld=ed. 
Then  Id  X  ±da=lda,  and  ed  X  ±(da+eb)  =beda,  %(eb+fc)  X 
fe=befc,   and   2/Xi/c=c/"9;  the  sum  of  all  which  will  be 
Io£>c21 ;  the  area  contained  between  the  stationary  line  1,  2 
and  the  boundary  Ia£c2. 

In  the  same  manner  you  may  find  the  area  of  ZihgZ,  of  j"£3i,j 
as  well  as  what  is  without  and  withinside  of  the  stationary 
line  7,  1. 

If  therefore  the  left-hand  offsets  exceed  the  right-hand  ones, 
it  is  plain  the  excess  must  be  added  to  the  area  within  the  sta- 
tionary lines ;  but  if  the  right-hand  offsets  exceed  the  left-hand 
ones  the  differencs  must  be  deducted  from  the  said  area,  if  the 


OF  OFFSETS. 

ground  be  kept  on  the  right-hand,  as  we  have  all  along  sup- 
posed ;  or  in  words  thus : 

To  find  the  contents  of  offsets. 

1.  From  the  distance  line  take  the  distance  to  the  preceding 
offset,  and  from  that  the  distance  of  the  one  preceding  it,  &c.  in 
four-pole  chains ;  so  will  you  have  the  respective  distances 
from  offset  to  offset,  but  in  a  retrograde  order. 

2.  Multiply  the  last  of  these  remainders  by  half  the  first 
offset,  the  next  by  half  the  sum  of  the  first  and  second,  the  next 
by  half  the  sum  of  the  second  and  third,  the  next  by  half  the 
sum  of  the  third  and  fourth,  &c.     The  sum  of  these  will  be  the 
area  produced  by  the  offsets. 

Thus,  in  the  foregoing  field-book  the  first  stationary  line  is 
22cA.  12/.,  or  llch.  12/.  of  four-pole  chains.     See  the  figure. 

ch.  L  ch.  L  ch.  L 

From  11. 12  =  1,  6.50  =  1/        3.90  =  le 

Take    6.50=1/          3.90=le         2.25=ld 

4.62=2/  2.60=e/         l.Q5=ed 


ch.l 

1<2=2.25  X  32?.,  half  the  first  offset,  .7200 

ed  =  1.65  X  IcA.  26Z.,  half  the  sum  of  the  1st  and  2d,  =  2.0790 
ef  =2.60  X  IcA.  32ZM  half  the  sum  of  the  2d  and  3d,  =  3.4320 
2/=4.62  X  37/.,  half  the  last  offset,  =  1.7094 

Contents  of  left  offsets  on  the  first  distance  in  ) 
square  four-pole  chains,  > 

In  like  manner  the  rest  are  performed. 

The  sum  of  the  left-hand  offsets  will  be  14.0856 

And  the  sum  of  the  right-hand  ones  3.6825 

Excess  of  left-hand  offets  in  sq.  four-pole  chains,     10.4031 

Acres  1.04031 


Perches  6.4496 

Excess  of  left-hand  offsets  above  the  right-hand  ones,  1  A. 
OR.  6P.,  to  be  added  to  the  area  within  the  stationary  lines. 


OF  OFFSETS. 


205 


SECTION  V. 

To  find  the  area  of  a  piece  of  ground  by  intersections  only,  when  all  the 
angles  of  the  field  can  be  seen  from  any  two  stations  on  the  outside  of  the 
ground. 

PL.  I*,  fig.  1. 

Let  ABCDEFG  be  a  field,  H  and  I  two  places  on  the  out- 
side of  it  from  whence  an  object  at  every  angle  of  the  field  may 
be  seen. 

Take  the  bearing  and  distance  between  H  and  /;  set  that 
at  the  head  of  your  field-book,  as  in  the  annexed  one.  Fix 
your  instrument  at  H,  from  whence  take  the  bearings  of  the 
several  angular  points  ABCD,  &c.  as  they  are  here  represented 
by  the  lines  HA,  HB,  HC,  HD,  <fec.  Again,  fix  your  instru- 
ment at  /  and  take  bearings  to  the  same  angular  points, 
represented  by  the  lines  I  A,  IB,  1C,  ID,  &c.,  and  let  the  first 
bearings  be  entered  in  the  second  column  and  the  second  bear- 
ings in  the  third  column  of  your  field-book  ;  then  it  is  plain 
that  the  points  of  intersection  made  from  the  bearings  in  the 
second  and  third  columns  of  every  line  will  be  the  angular 
points  of  the  field,  or  the  points  A,  B,  C,  D,  &c.,  which  points 
being  joined  by  right  lines  will  give  the  plan  ABCDEFG  A 
required. 


Bear.  180,  Dist.  28cA.  of  the  Sta.  Hand  /. 


No. 

Bear. 

Bear. 

A 
B 
C 
D 
E 
F 
G 

26H 
265f 
248 
2381 
2151 
2081 
220" 

3311 
3171 
3071 
289 
2621 
2861 
300" 

The  same  may  be  done  from  any  two  stations  withinside  of 
the  land  from  whence  all  the  angles  of  the  field  can  be  seen. 

This  method  will  be  found  useful  in  case  the  stationary  dis- 
tances from  any  cause  prove  inaccessible,  or  should  it  be  re- 
quired to  be  done  by  one  party  when  the  other,  in  whose  pos- 
session it  is,  refuses  to  admit  you  to  go  on  the  land. 


206  BY  INTERSECTIONS. 

I 

To  find  the  contents  of  a  field  by  calculation,  which  was  taken  by  inter' 
section. 

In  the  triangle  AIH,  the  angles  AHI,  AIH,  and  the  base' 
HI  being  known,  the  perpendicular  Aa  and  the  segments  of  the 
base  //a,  AI  may  be  obtained  by  trigonometry :  and  in  the 
same  manner  all  the  other  perpendiculars,  Bb,  Cc,  Dd,  Ee,  Ff, 
Gg,  and  the  several  segments  at  b,  c,  d,  e,f,g\  if,  therefore,  the 
several  perpendiculars  be  supposed  to  be  drawn  into  the  scheme 
(which  are  here  omitted,  to  prevent  confusion  arising  from  a 
multiplicity  of  lines),  it  is  plain  that  if  from  bBCDEeb  there  be 
taken  bBAGFeb  the  remainder  will  be  the  map  ABCDEFGA. 

As  before,  half  the  sum  of  .B^and  Cc  multiplied  by  be  will  be 
the  area  of  the  trapezium  bBCc-,  after  the  same  manner,  half 
the  sum  of  Cc  and  Dd  multiplied  by  cd  will  give  the  area  of  the 
trapezium  cCDd ;  and  again,  half  the  sum  of  Dd  and  Ee  mul- 
tiplied by  de  gives  the  area  of  the  trapezium  dDEe  ;  and  the 
sum  of  these  three  trapezia  will  be  the  area  of  the  figure 
bBCDEeb.  , 

Again,  in  the  same  manner,  half  the  sum  of  Bb  and  Aa  mul 
tiplied  by  ab  will  give  the  area  of  the  trapezium  BbAa,  and 
half  the  sum  of  a  A  and^O  by  ag  gives  the  trapezium  aAGg\ 
to  thes6  add  the  trapezia  gGFfand  fFEe,  which  are  found  in 
the  like  manner,  and  you  will  have  the  figure  bBAGFEeb,  and 
this  taken  from  bBCDEeb  will  leave  the  map  ABCDEFGA. 
Q.  E.  I. 

It  will  be  sufficient  to  protract  this  kind  of  work,  and  from 
the  map  to  determine  the  area  as  well  as  in  plate  10,  fig.  3,  to 
find  the  areas  of  the  pieces  3,  4,  5,  6,  3  and  6,  7,  7,  6  from 
geometrical  constructions. 

How  to  determine  the  station  where  a  fault  has  been  committed  in  afield- 
book,  without  the  trouble  of  going  round  the  whole  ground  a  second  time. 

From  every  fourth  or  fifth  station,  if  they  be  not  very  long 
ones,  or  oftener  if  they  are,  let  an  intersection  be  taken  to  any 
object,  as  to  any  particular  part  of  a  castle,  house,  or  cock  of 
hay,  &c.,  or,  if  all  these  be  wanting,  to  a  long  staff  with  a 
white  sheet  or  napkin  set  thereon,  to  render  the  object  more 
conspicuous,  and  let  this  be  placed  on  the  summit  of  the.  land, 
and  let  the  respective  intersections  so  taken  be  inserted  on  the 
left-hand  side  of  the  field-book  opposite  to  the  stations  from 
whence  they  were  respectively  taken. 

In  your  protraction  as  you  proceed  let  every  intersection  be 
laid  off  from  the  respective  stations  from  whence  they  were 
taken,  and  let  these  lines  be  continued ;  if  they  all  converge  or 


BY  INTERSECTIONS,  207 

meet  in  one  point,  we  thence  conclude  all  is  right,  or  so  far  a» 
they  do  converge;  but  if  we  find  a  line  of  intersection  to 'di- 
verge or  fly  off  from  the  rest,  we  may  be  sure  that  either  a  mis- 
take has  happened  between  the  station  the  foregoing  intersec- 
tion was  taken  at  and  the  station  from  whence  the  intersection 
line  diverges,  or  there  must  be  an  error  in  the  intersection ;  but 
to  be  assured  in  which  of  these  the  fault  is,  protract  on  to  the 
next  intersection,  and  having  set  it  off,  if  it  converges  with  the 
rest,  though  the  foregoing  one  did  not,  we  may  conclude  the 
fault  was  committed  hi  taking  the  last  intersection  but  one,  and 
none  in  any  station,  and  that  so  far  is  true  as  is  protracted  \ 
but  if  this  as  well  as  the  foregoing  intersection  diverge  or  fly 
from  the  point  of  concourse  or  converging  point  of  the  rest,  the 
error  must  have  its  rise  from  some  station  or  stations  at  or  after 
that  from  whence  the  last  converging  intersection  line  was 
taken :  so  that  by  going  to  that  station  on  the  ground,  and  pro- 
ceeding on  to  that  where  the  next  or  from  whence  the  follow- 
ing diverging  intersection  was  taken,  we  can  readily  and  with 
little  trouble  set  all  to  rights. 

But  in  most  tracts  of  land  one  object  cannot  be  seen  from 
every  station,  or  from  perhaps  one-fourth  of  them ;  in  this  case 
we  are  under  the  necessity  to  move  the  pole  after  we  begin  to 
lose  sight  of  it,  to  some  other  part  of  the  land,  where  it  may 
be  seen  from  as  many  more  stations  as  possible ;  which  is 
easily  done  by  viewing  the  boundary  before  it  be  surveyed : 
the  pole  then  being  fixed  in  an  advantageous  place,  the  first 
intersection  to  it  is  best  to  be  made  from  the  same  station  from 
whence  the  last  one  was  taken,  and  then  as  often  as  may  be 
thought  convenient,  as  before ;  in  like  manner  the  whole  may 
be  done  by  the  removal  of  the  pole. 

When  we  here  speak  of  stations,  we  do  not  mean  such  as 
are  usually  taken  at  every  particular  angle  of  the  field :  for  it 
is  to  be  apprehended,  that  every  skilful  surveyor,  particularly 
such  who  use  calculation,  will  take  the  longest  distances  pos- 
sible, not  only  to  lessen  the  number  of  stations,  for  the  ease  of 
either  protraction  or  calculation,  but  with  greater  certainty  to 
account  for  the  land  passed  by,  on  the  right-hand  or  on  the  left, 
which  is  taken  by  offsets :  and  surely  it  will  be  allowed  that 
any  measure  taken  on  the  ground,  and  the  contents  thence  arith- 
metically computed,  will  be  much  more  accurate  than  that 
which  is  obtained  from  any  geometrical  projection. 

From  what  has  been  said  it  is  plain,  that  from  this  method 
any  fault  committed  in  a  survey  can  be  readily  determined,  and 
therefore  must  be  much  preferable  to  the  present  method  of 
taking  diagonals,  or  the  bearings  and  lengths  of  lines  across 


808         TO  ENLARGE  OR  DIMINISH  MAPS. 

land,  to  accomplish  that  end ;  which  last  method  i's  too  fre- 
quently used  by  surveyors  to  approximate  or  arrive  near  the 
contents,  which  will  ever  remain  uncertain,  let  these  diagonals  be 
ever  so  many,  till  the  station  or  stations  wherein  the  error  or 
errors  were  committed  be  found;  and  the  fault  or  faults  be 
corrected. 

Where  one  diagonal  is  taken,  it  may  perhaps  close  or  meet 
with  one  part  of  the  survey  and  not  with  the  other ;  in  this 
case,  if  the  surveyor  would  discover  his  error,  he  must  survey 
that  part  of  the  land  which  did  not  close,  and  this  may  be  half 
or  more  of  the  whole.  And  should  the  diagonal  close  with 
neither  part,  but  be  too  long  or  too  short,  or  should  it  fall  on 
either  side  of  the  assigned  point  it  was  to  close  with,  he  ought 
to  go  over  the  whole,  and  make  a  new  survey  of  it,  in  order  to 
discover  his  error. 

A  number  of  diagonals  are  frequently  taken,  the  sum  of  the 
lengths  of  which  very  often  exceeds  the  circuit  of  the  ground, 
and  after  all  they  are  but  approximations,  and  the  contents 
remain  uncertain  as  before  ;  therefore,  he  who  returns  a  map 
made  up  by  the  assistance  of  diagonals,  where  there  remains  a 
misclosure  in  any  one  part,  runs  the  risk  of  being  detected  in 
an  error,  and  must  suffer  uneasiness  in  his  mind,  as  he  cannot 
be  certain  of  the  return  he  makes. 

The  frequent  misclosures  which  are  botched  up  by  diagonals 
occasion  the  many  and  frequent  scandalous  broils  and  animosi- 
ties between  surveyors,  which  tend  to  the  loss  of  character  of 
the  one  or  the  other,  and  indeed  often  to  the  disrepute  of  both, 
as  well  as  to  that  of  the  science  they  profess. 

But  these  may  be  easily  remedied  by  intersections  and  the 
bearing  or  line  to  be  adjusted  where  the  fault  was  committed  ; 
and  till  this  be  found,  nothing  can  be  certain. 


SECTION  VI. 
TO  ENLARGE  OR  DIMINISH  MAPS. 

To  enlarge  or  diminish  a  map,  or  to  reduce  a  map  from  one  scale  to 
another  ;  also,  the  manner  of  uniting  separate  maps  of  lands  which  join  each 
Other  into  one  map  of  any  assigned  size. 

LAY  the  map  you  would  enlarge  over  the  paper  on  which 
you  would  enlarge  it,  and  with  a  fine  protracting  pin  prick 
through  every  angular  point  of  your  map  ;  join  these  points  on 
your  paper  (laying  the  map  you  copy  before  you)  by  pencilled 


TO  ENLARGE  OR  DIMINISH  MAPS.        209 

or  popped  lines,  and  you  have  the  copy  of  the  map  you  are  to 
enlarge:  in  this  manner  any  protraction  may  be  copied  on 
paper,  vellum,  or  parchment,  for  a  fair  map. 

If  you  would  enlarge  a  map  to  a  scale  which  is  double,  or 
treble,  or  quadruple  to  that  of  the  map  to  be  enlarged,  the 
paper  you  must  provide  for  its  enlargement  must  be  two,  or 
three,  or  four  times  as  long  and  broad  as  the  map ;  for  which 
purpose  in  large  things  you  will  find  it  necessary  to  join  several 
sheets  of  paper,  and  to  cement  them  with  white  wafer  or  paste, 
but  the  former  is  best. 

Then  pitch  upon  any  point  in  your  copied  map  for  a  centre ; 
from  whence  if  distances  be  taken  to  its  extreme  points,  and 
thence  if  those  distances  be  set  in  a  right  line  with  (but  from) 
the  centre,  and  these  last  points  fall  within  your  paper,  the  map 
may  be  increased  on  it  to  a  scale  as  large  again  as  its  own ;  and 
if  the  like  distances  be  again  set  outwards  in  right  lines  from 
the  centre,  and  if  these  last  points  fall  within  your  paper,  it  will 
contain  a  map  increased  to  a  scale  three  tunes  as  large  as  its 
own,  <fec. 

PL.  12.  Jig.  2. 

Let  the  pricked  or  popped  lines  represent  the  copy  of  a  down 
or  old  survey,  laid  down  by  a  scale  of  80  perches  to  an  inch, 
and  let  it  be  required  to  enlarge  it  to  one  laid  down  by  40  to 
an  inch. 

Pitch  upon  your  centre  as  © ,  from  whence  through  a  lay  the 
fiducial  edge  of  a  thin  ruler ;  with  a  fine-pointed  pah*  of  com- 
passes take  the  distance  from  a  to  the  centre  © ,  and  lay  it  by 
the  ruler's  edge  from  a  to  A ;  in  the  like  manner  take  the  dis- 
tance from  the  next  station  b  to  the  centre  ©  ,  and  lay  it  over  in 
a  right  line  from  b  to  .B,  and  join  the  points  A  and  B  by  the 
right  line  AB ;  in  the  like  manner  set  over  the  distance  from 
every  station  to  the  centre,  from  that  station  outwards,  and  you 
will  have  every  point  to  enlarge  to:  the  joining  of  these  con- 
stantly as  you  go  on  by  right  lines  will  give  you  the  enlarged 
map  required. 

In  taking  the  distance  from  every  station  to  the  centre,  set 
one  foot  of  the  compasses  in  the  station,  and  the  other  very 
lightly  over  the  centre-point,  so  lightly  as  scarcely  to  touch  it, 
otherwise  the  centre-point  will  become  so  wide,  that  it  may 
occasion  several  er/ors  in  the  enlarged  map :  for  if  you  err 
from  the  exact  centre  but  a  little,  that  error  will  become  double, 
or  treble,  or  quadruple,  as  you  enlarge  to  a  scale  that  is  double, 
or  treble,  or  quadruple  of  the  given  one ;  therefore  great  accu- 
racy is  required  in  enlarging  a  map. 


210        TO  ENLARGE  OR  DIMINISH  MAPS. 

When  you  have  done  with  a  station,  give  a  dash  with  a  pea 
or  pencil  to  it,  such  as  at  the  station  a  and  b :  by  this  means 
you  cannot  be  disappointed  in  missing  a  station,  or  in  laying 
your  ruler  over  one  station  twice. 

From  what  has  been  said  it  is  plain,  that  if  a  map  is  to  be 
enlarged  to  one  whose  scale  is  double  the  given  one,  that  the 
distances  from  the  respective  stations  to  the  centre,  being  set 
over  by  the  ruler's  edge,  will  give  the  points  fpr  the  enlarged  one. 
And  thus  may  a  map  be  enlarged  from  a  scale  of  160  to  one  of 
80,  from  one  of  80  to  one  of  40,  from  one  of  20  to  one  of  10 
perches  to  an  inch,  &c.  For  to  enlarge  to  a  scale  that  is  double, 
the  number  of  perches  to  an  inch  for  the  enlarged  map  must  be 
half  of  those  to  an  inch  for  that  to  be  enlarged :  to  enlarge  to 
a  scale  that  is  treble  the  given  one,  the  number  of  perches  to 
an  inch  for  the  enlarged  map  will  be  one-third  of  those  for  the 
other ;  if  to  a  scale  that  is  quadruple  the  given  one,  the  number 
of  perches  to  an  inch  for  the  enlarged  map  will  be  one-fourth 
of  those  for  the  other,  &c. :  therefore,  if  you  would  enlarge  a 
map  which  is  laid  down  by  a  scale  of  120  perches  to  an  inch 
to  one  of  40  perches  to  an  inch,  the  distance  from  the  several 
stations  to  the  centre,  being  set  twice  beyond  the  said  stations, 
will  mark  out  the  several  points  required,  for  these  points  will 
be  three  times  farther  from  the  centre  than  the  stationary  points 
of  the  map  are. 

In  the  same  manner,  if  you  would  enlarge  a  map  from  a 
scale  of  160  to  one  of  40  perches  to  an  inch,  the  distance  from 
the  several  stations  to  the  centre,  being  set  three  times  beyond 
said  stations,  will  lay  out  the  points  for  your  enlarged  map,  for 
these  points  will  be  four  times  farther  from  the  centre  than  are 
the  stations  of  the  map. 

When  a  map  is  enlarged  to  another,  whose  scale  is  double, 
or  treble,  or  quadruple,  &c.  of  the  given  one,  every  line,  as 
well  as  the  length  and  breadth  of  the  enlarged  map,  will  be 
double,  or  treble,  or  quadruple,  &c.  those  of  the  given  one,  for 
it  must  be  easy  to  conceive  that  those  maps  are  like :  but  the 
area  if  the  scale  be  double  will  be  four  times,  if  treble  nine 
times,  if  quadruple  sixteen  times  that  of  the  given  figure; 
that  is,  it  will  contain  four,  nine,  or  sixteen  times  as  many  square 
inches  as  the  given  one  (for  it  has  been  shown  that  like  polygons 
are  in  a  duplicate  proportion  with  the  homologous  sides).  Yet 
these  figures  being  cast  up  by  their  respective  scales  will  pro- 
duce the  same  contents. 

Thus  much  is  sufficient  for  enlarging  maps,  and  from  hence, 
diminishing  of  them  will  be  obvious ;  for  one-fourth,  one-third, 
or  half  the  distances  from  the  several  stations  to  the  centre 


TO  ENLARGE  OR  DIMINISH  MAPS.         211 

will  mark  out  points  which  if  joined  will  compose  a  map 
similar  to  the  given  one,  whose  scale  will  be  four  times,  three 
times,  or  twice  as  small  as  the  given  one. 

Thus,  if  we  would  reduce  a  map  of  from  40  to  80,  from  20 
to  40,  from  10  to  20  perches  to  an  inch,  &e.,  half  the  distance 
of  the  stations  from  the  centre  will  give  the  points  requisite  for 
drawing  the  map  ;  if  we  would  reduce  from  40  to  120,  from  20 
to  60,  from  10  to  30  perches  to  an  inch,  &c.,  one-third  of  the 
distances  to  the  centre  will  give  the  points  for  the  map ;  and  if 
we  would  reduce  from  40  to  160,  from  20  to  80,  from  10  to  40 
perches  to  an  inch,  &c.,  one-fourth  of  the  distances  to  the  centre 
will  give  the  points  for  the  map. 

By  the  methods  here  laid  down  I  have  reduced  a  map  from 
a  scale  of  40  to  one  of  20  perches  to  an  inch,  which  contained 
upwards  of  1200  acres,  and  consisted  of  224  separate  divisions, 
without  the  least  confusion  from  the  lines ;  for  none  can  arise 
if  the  methods  here  laid  down  be  strictly  observed. 

I  have  also  from  the  same  methods  reduced  a  large  book  of 
maps,  each  of  which  was  an  entire  skin  of  parchment,  and  the 
whole  contained  upwards  of  46,000  acres,  to  a  pocket  volume  ; 
and  afterward  connected  all  these  maps  into  one  map,  which 
was  contained  in  one  skin  of  parchment :  therefore,  upon  the 
whole  I  do  recommend  these  methods  for  reducing  maps  to  be 
much  more  accurate  than  any  of  the  methods  commonly  used, 
such  as  squaring  of  paper,  using  a  parallelogram,  proportion- 
able compasses,  or  any  other  method  I  ever  met  with,  though 
the  figures  to  be  reduced  were  ever  so  numerous,  irregular,  or 
complicated. 

To  unite  separate  maps  of  lands  which  join  each  other  into  one  map  of  any 
assigned  size. 

If  there  be  several  large  maps  contained  in  a  book,  each  of 
which  suppose  to  take  up  a  skin  of  parchment,  or  a  sheet  of 
the  largest  paper,  which  maps  of  lands  join  each  other,  -and 
it  be  required  to  reduce  them  to  so  small  a  scale  that  all  of 
them  when  joined  together  may  be  contained  in  one  skin,  half  a 
skin,  or  any  assigned  sized  piece  of  parchment  or  paper : — 

Having  pricked  off  and  copied  the  several  maps  on  any  kind 
of  paper,  unite  them  by  cutting  with  scissors  along  the  edge 
of  one  boundary  which  is  adjoining  the  other,  but  not  cutting 
by  the  edge  of  both,  and  throw  aside  the  parts  cut  off:  then 
lay  these  together  on  a  large  table,  or  on  the  floor,  and  where  the 
boundaries  agree  they  will  fit  in  with  each  other  as  indentures 
do  ;  and  after  this  manner  they  are  easily  connected :  measure 
then  the  length  ana  breadth  of  the  entire  connected  maps,  and 


ft 

212         TO  ENLARGE  OR  DIMINISH  MAPS. 

the  length  and  breadth  of  the  parchment  or  paper  you  are  con- 
fined to  ;  if  the  former  be  three,  four,  or  five  times  greater  (that 
'  is,  longer  and  broader)  than  the  latter,  reduce  each  copied  map 
severally  to  a  scale  that  is  three,  or  four,  or  five  times  less,  as 
before ;  and  the  same  parts  of  the  boundaries  you  cut  by  in  the 
large  maps,  by  the  same  you  must  also  cut  in  small  ones,  and 
unite  the  small  as  the  large  ones  were  united  ;  cementing  them 
together  with  white  wafer :  thus  your  map  will  be  reduced  to  the 
assigned  size,  which  copy  over  fair  on  the  parchment  or  paper 
you  were  confined  to. 

But  it  is  not  always  that  a  person  is  confined  to  a  given  area 
of  parchment,  or  paper  ;  in  such  cases,  if  there  are  many  large 
maps  to  be  united  into  one,  reduce  each  of  them  severally  to  a 
scale  of  160  perches  to  an  inch,  and  unite  those  by  the  con* 
tiguity  or  boundaries,  as  before :  or  if  you  have  a  few,  it  will 
be  sufficient  to  reduce  them  to  a  scale  of  120,  &c.  But  having 
the  maps  given,  and  the  scale  by  which  they  are  laid  down, 
your  reason  will  be  sufficient  to  direct  you  to  know  what  scale 
they  should  be  reduced  to. 

Directions  concerning  surveys  in  general. 

If  you  have  a  large  quantity  of  ground  to  survey,  which  con- 
sists of  many  fields  or  holdings,  and  that  it  be  required  to  map 
and  give  the  respective  contents  of  the  same,  it  is  best  to  make 
a  survey  of  the  whole  first,  and  to  be  satisfied  that  it  is  truly 
taken,  as  well  as  to  find  its  contents  ;  and  as  you  go  round  the 
land,  to  make  a  note  on  the  side  of  your  field-book  at  every 
station  where  -the  boundary  of  any  particular  field  or  holding 
intersects  or  meets  the  surround ;  then  proceed  from  any  one 
of  those  stations,  and  in  your  field-book  say,  "  proceed  from 
such  a  station,"  and  when  you  have  gone  round  that  field  or 
division,  insert  the  station  you  close  at,  and  so  through  the 
whole  :  a  little  practice  only  can  render  this  sufficiently  familiar, 
and  the  method  of  protraction  must  be  evident  from  the  field- 
notes.  When  the  whole  is  protracted,  and  you  are  satisfied  of 
the  closes  of  the  particular  divisions,  cast  up  each  severally, 
and  if  the  sum  of  their  contents  be  equal  to  the  contents  of  the 
whole  first  found,  you  may  safely  conclude  that  all. is  right. 

The  protraction  being  thus  finished  and  cast  up,  transfer  it 
on  clean  paper,  vellum,  or  parchment  as  before  ;  be  careful  to 
draw  your  lines  with  a  fine  pen,  write  on  it  the  names  of  the 
circumjacent  lands,  and  set  No.  1,  2,  3,  4,  &c.  in  every  par- 
ticular field  or  division  ;  let  every  tenant's  particular  holding  be 
distinguished  by  a  different  coloured  paint  being  run  finely  along 
the  boundaries  ;  let  all  the  roads,  rivulets,  rivers,  bogs,  ponds, 


DIVISION  OF  LAND.  213 

houses,  castles,  churches,  beacons,  or  whatever  else  may  be 
remarkable  on  the  ground,  be  distinguished  on  the  map. 
Write  the  title  of  the  map  in  a  neat  compartment,  either  drawn  or 
done  from  a  good  copper-plate  engraving,  with  the  gentleman's 
arms.  Prick  off  one  of  your  parallels  with  the  map,  and  on  it 
make  a  mariner's  compass,  and  draw  a  flower-de-luce  to  the 
north,  and  this  will  represent  the  magnetical  north ;  after  which 
set  off  the  variation,  which  express  in  figures,  and  through  the 
centre  of  the  compass  let  a  true  meridian  line  be  drawn  of  about 
3  inches  long,  by  which  write  True  Meridian.  Let  a  scale  be 
drawn,  or  it  is  sufficient  to  express  the  number  of  perches  to 
an  inch  the  map  was  laid  down  by.  Draw  a  reference  table 
of  three  or,  if  occasion  be,  of  four  or  more  columns :  in  the 
first  insert  the  number  of  the  field  or  holding ;  in  the  next  its 
name,  and  by  whom  occupied ;  in  the  third  the  quantity  of 
acres,  roods,  and  perches  it  contains  ;  if  you  have  unprofitable 
land,  as  bog  or  mountain,  let  the  quantity  be  inserted  in  the 
fourth  column ;  and  if  it  be  required,  you  may  make  another 
column  for  statute  measure,  and  then  the  map  is  completed. 


SECTION  VII. 

THE  METHOD  OF  DIVIDING  LAND,  OR  OF  TAKING 
OFF  OR  ENCLOSING  ANY  GIVEN  QUANTITY. 

EXAMPLE    I. 
PL.  12.  fig.,1. 

LET  ABCD,  &c.  be  a  map  of  ground  containing  11  acres ; 
it  is  required  to  cut  off  a  piece,  as  DEFGID,  that  shall  con- 
tain 5  acres. 

Join  any  two  opposite  stations,  as  D  and  6r,  with  the  line 
DG  (which  you  may  nearly  judge  to*be  the  partition  line),  and 
find  the  area  of  the  part  DEFG,  which  suppose  may  want  3R. 
20P.  of  the  quantity  you  would  cut  off;  measure  the  line  DG, 
which  suppose  to  be  70  perches.  Divide  3R.  20P.,  or  140J*., 
by  25,  the  half  of  D6r,  and  the  quotient  4  will  be  a  perpendicular 
for  a  triangle  whose  base  is  70  and  area  140P.  Let  HI  be 
drawn  parallel  to  DG,  at  the  distance  of  the  perpendicular  4, 
and  from  /,  where  it  cuts  the  boundary,  draw  a  line  to  D,  and 
that  line  DI  will  be  the  division  line  ;  or  a  line  from  G  to  H 


DIVISION  OF  LAND. 

will  have  the  same  effect ;  all  which  must  be  evident  from  what 
has  been  already  said. 

But  if  hills,  trees,  &c.  obstruct  the  view  of  the  points  D  and 
7  from  each  other,  it  will  be  necessary,  in  order  to  run  a  par- 
tition line,  to  know  its  bearing ;  and  it  may  be  proper  on  some 
occasions  to  have  its  length ;  both  these  may  be  easily  calcu- 
lated from  the  common  field-notes  only,  as  in  the  following  ex- 
ample, without  the  trouble  of  any  other  measurement  on  the 
ground,  or  any  dependence  on  the  map  and  scale. 

EXAMPLE    II. 

PL.  12.  Jig.  3. 

Let  ABCDEFGHIA  be  a  tract  of  land  to  be  divided  into 
two  equal  parts  by  a  right  line  from  the  corner  /  to  the  oppo- 
site boundary  CD ;  required  the  bearing  and  length  of  the  par- 
tition line  IN,  by  calculation,  from  the  following  field-notes,  viz. 


Field-Notes  and  Area. 

Bound. 

Bearing. 

Perch. 

AB 

N  19°    0'  E 

108 

BC 

S77     0  E 

91 

CD 

S  27     0  E 

115 

DE 

S  52     0  W 

58 

EF 

S  15   30  E 

76 

FG 

West. 

70.9 

GH 

N36     0  W 

47 

HI 

North. 

64.3 

IA 

N  62    15  W 

59 

152A.  1R.  25.9P. 

Operation. 


IABCI 

Per. 

N. 

S. 

E. 

W. 

s 

IA 

N    621°  W 

59 

27.5 



___ 

52.2 

Ei 

AB 

N    19       E 

188 

102.1 



35.2 



BC 

S    77      E 

91 

20.5 

88.7 



fr 

CI 

__    . 

— 

— 

109.1 

— 

71.7 

g» 

o 

Area,  8722.3  perches. 

129.6 

129.6 

123.9 

123.9 

DIVISION  OF  LAND. 

152A.  1R.  25.9P.         =24385.9  perches. 
Half,  to  be  divided  off,  =12192.9  i      , 
The  part  1ABCI         =  8722.3  J  Sl 


215 


Triangle  ICNI 


=  3470.6  perches. 


ICDI. 

Per. 

N. 

S. 

E. 

W. 

3 

1C 

N  —  E 

115 

109.1 



71.9 

— 

§' 

CD 

S  27  E 

— 

— 

102.5 

52 

— 

DI 

6.6 

— 

123.9 

1 

Area  6522.1  per. 

109.1 

109.1 

123.9 

123.9 

p 

„-  (  ICDI  :  CD:  :  ICNI  :  CN 

Then,  as  j  ^.i  .  n?  .  .  ^  6  .  6Li 

which  determines  the  point  N  in  CD. 


..        ia 
theo.  18,sec.l, 


ICNI. 

Per. 

N. 

S. 

E. 

W. 

1C 

CN 
NI 

as  before 

S27°E 

61.2 

109.1 

54.6 
54.6 

71.7 

27.8 

99.5 

As  dif.  lat.  54.6 

:   Radius     S.  90° 

:  :  Depart.  99.5 

:  Tang.  Bear.  61°  15' 


As  S.  Bear.     61°  15' 
:  Depart.         99.5 
:  :  Radius  S.  90° 
:  Distance     113.49 


In  the  part  I  ABC  I,  the  difference  between  the  northings  and 
the  southings  of  the  three  lines  I  A,  AB,  and  BC  (109.1)  is 
the  difference  of  latitude,  and  that  of  their  eastings  and 
westings  (71.7)  the  departure  of  the  line  C/,  which  is  placed 
thereto,  so  as  to  balance  the  columns;  see  theo.  1,  sect  5: 
hence  the  contents  are  obtained,  as  already  taught,  without  the 
bearing  or  length  of  the  line  CI. 

For  the  triangle  ICDI,  the  dif.  lat.  and  dep.  of  1C  are 
taken  from  the  preceding  table,  which  in  going  from  /  to  C 


216  DIVISION  OF  LAND. 

will  be  northing  and  easting :  those  of  CD  are  found  by  the 
bearing  and  distance,  and  of  DI  by  balancing  the  columns,  as 
before  for  CI. 

The  difference  of  latitude  (54.6)  and  departure  (99.5)  of  the 
line  NI,  in  the  third  table,  are  found  by  balancing  those  of  1C 
and  CJV;  and  as  they  are  the  base  and  perpendicular  of  a 
right-angled  triangle,  of  which  the  line  NI  is  the  hypothenuse, 
and  the  angle  opposite  to  the  departure  the  bearing,  we  have 
the  answer  by  two  trigonometrical  statings,'  as  above  ;  and  thus 
may  any  tract  be  accurately  divided,  or  any  proposed  quantity 
readily  cut  off  or  enclosed. 

Now  the  student  or  practitioner  may  calculate  the  contents  of 
the  part  ABCNIA  (the  bearing  and  distance,  or  the  dif.  lat. 
and  dep.  of  CN  and  of  NI  being  known),  and  if  it  be  found 
equal  to  the  intended  quantity,  it  proves  the  truth  of  the 
operation. 

EXAMPLE    III. 
PL.  12.  fig.  3. 

It  is  proposed  to  cut  off  38A.  16|P.  to  the  south  end  of  this 
tract,  by  a  line  running  from  E  due  west  40  perches  to  a  well 
at  O,  and  from  thence  a  right  line  to  a  point  M  in  the  boundary 
HI ;  the  place  of  M  and  the  bearing  and  length  of  the  line 
OM  are  required,  the  field-notes  being  as  in  example  second. 

%  M  from  H,  north  43.23  ) 
Answer,  J  Q^  ^  7g0  r  w  3g  Q3  J  perches. 


In  this  example  we  find, 

The  area  of  OEFGHO          =5270.5  \ 
Consequently  of  HOMH        =  826.0  f         , 
Dif.  lat.  of  the  MneHO=HV  =     35.2  (  pei 
Departure  of  ditto  •  =  O  V        =     38.2  ) 

As  HI  happens  to  be  a  meridian,  the  area  of  HOMH  divided 
by  half  OV  (19.1)  quotes  HM  (42.23),  without  finding  the 
area  of  HOIH,  as  we  did  of  ICDI  in  example  second,  and 
HM—HV=VM=8.03—  dif.  lat.  of  OM,  which  with  its  dep. 
FO=38.2,  gives  the  bearing  and  distance  as  before. 

EXAMPLE    IV. 
PL.  12.  fa.  4. 


A  trapezoidal  field  ABCD,  bounded  as  under  specified,  is  to 
be  divided  into  two  equal  parts  by  a  right  line  EF  parallel  to 
AB  or  CD ;  required  AF  or  BF. 


DIVISION  OF  LAND. 


317 


Bou. 

Bearing. 

Per. 

AB 
BC 
CD 
DA 

South. 

N80°  W 
N  39£  W 
S  80    E 

30 
60 
45.5 
89.4 

13  A.  3R.  7P. 

In  the  triangle  CBG  are  given  BC  and  all  the  angles 
(known  by  the  bearings)  to  find  J5Gf,  and  thence  the  area  by 
prob.  9,  sect.  4.  which,  +  half  the  area  of  ABCD  =  area  of 
EFG  ;  then,  as  the  area  of  CBG  is  to  that  of  EFG,  so  is  the 
square  of  BG  to  the  square  of  FG,  andjFG — BG=BF 

Operation  at  large. 

Angle  G  39°  30',  log.  S.  Co.  Ar.  0.19649) 
Side  BC  60  per.  log.  1.77815  >  add 

Angle  C40°  30',  sine  9.81254  ) 


Side  BG  61.26  per. 
Side  BC  60  per. 
Angle  B  100°  0',  sine 

2)3619.8,  log. 
As  CBG=  1809.9  Co.  Ar. 


1.78718 
1.77815  > add 
9.99335 

3.55868 


Isto  jEjPG?=  2913.4,  log. 
Soissq.BG61.26,  log. 

To  sq.  FG  77.72, 

Ans.  BF=  16.46  perches 


6.74235 

3.46440 

1.78718 
1.78718 

2)3.78111 
1.89055 


add 


By  the  application  of  this  method  a  tract  of  land  may  be 
divided  accurately,  in  any  proportion,  by  a  line  running  in  any 
assigned  direction. 

Note. — When  the  practitioner  would  wish  to  be  very  accu- 
rate, it  will  be  much  better  to  work  by  four-pole  chains  and 
links  than  by  perches  and  tenths ;  one-tenth  of  a  perch  square 
being  equal  to  6^  square  links. 


'218 


DIVISION  OF  LAND. 


EXAMPLE  V. 


The  following  Field-notes  (from  A.  BURNS)  are  of  a  piece  of 
land,  which  is  proposed,  as  an  example,  to  be  divided  into  three 
equal  parts  by  two  right  lines  running  from  the  sixth  and  seventh 
stations :  and  proved  by  calculating  the  contents  of  the  middle  part. 


St. 

Bearing. 

4-P.  eh. 

1 

N.E.  561° 

21.60 

2 

N.E.  26i 

13.44 

3 

S.E.  711 

18.96 

4 

S.E.  261 

13.44 

5 

S.W.  711 

18.96 

6 

S.E.  45 

8.47 

7 

S.E.  63i 

13.44 

8 

N.E.  45 

8.47 

9 

S.E.  261 

13.44 

10 

S.W.  45 

8.47 

11 

S.W.  63£ 

13.44 

12 

N.W.  76 

24.73 

13 

N.W.  36| 

30.00 

Area,  167A.  1R.  24P. 

EXAMPLE    VI. 
PL.  8.  Jig.  5. 

The  plot  ABCDEFGHA  is  proposed  to  be  divided  geometri- 
cally, in  the  proportion  of  2  to  3,  by  a  right  line  from  a  given 
point  in  any  boundary  or  angle  thereof,  suppose  the  point  D. 

Reduce  the  plot  to  the  triangle  cDe,  as  already  taught; 
divide  the  base  ce  in  the  point  N,  so  that  cN  be  to  Ne  in  the 
ratio  of  2  to  3,  by  prob.  14  ;  draw  DN,  and  it  is  done. 


DIVISION  OF  LAND.  219 

EXAMPLE  VH. 
PL.  12.  fig.  3. 

Example  second  may  likewise  be  performed,  geometrically. 
Produce  CD  both  ways  for  a  base,  and  reduce  the  whole  to 
a  triangle,  making  /  the  vertical  point ;  then  bisect  the  base  in 
N,  and  draw  IN.     But, 

Notwithstanding  this  geometrical  method  is  demonstrably 
true  in  theory,  it  is  not  as  safe,  on  practical  occasions  requiring 
accuracy,  as  the  calculation,  even  when  performed  with  the 
greatest  care ;  for  which  reason  we  will  not  enlarge  on  it  here. 

EXAMPLE  vm. 

Suppose  864  acres  to  be  laid  out  in  form  of  a  right-angled  parallelogram^ 
of  which  the  sides  shall  be  in  proportion  as  5  to  3 ;  required  their  dimensions. 

For  the  greater  side,  multiply  the  area  by  the  greater  num- 
ber of  the  given  proportion,  and  divide  by  the  less,  or,  for  the 
less  side,  multiply  by  the  less  number,  and^ivide  by  the  greater ; 
the  square  root  of  the  quotient  will  be  the  side  required :  thus, 
864A.=  138240P.  1.38240 

5  3 

3)691200  5)414720 

Ans.  V  230400=480.          V  82944=288 

EXAMPLE  IX. 

If  it  be  required  to  lay  out  any  quantity  of  ground,  suppose 
47 A.  2R.  16P.,  in  form  of  a  parallelogram,  of  which  the 
length  is  to  exceed  the  breadth  by  a  given  difference,  for  instance, 
80  perches,  then  add  the  square  of  half  this  difference  to  the 
area,  and  take  the  square  root  of  the  sum ;  to  which  add  half 
the  difference  for  the  greater  side,  and  subtract  it  therefrom  for 
the  less :  thus, 

47A.  2R.  16P.=7616  perches, 
1600 


V  9216=96 
1600  half  diff. ;      add  and  subt.  40 


( the  length =136 
( the  breadth=56 


Ans 

K2 


220  OF  SURVEYING  HARBOURS,  &c. 

Any  proposed  quantity  of  ground  may  be  laid  out  or  enclosed 
in  the  form 

Square .     .     by  prob.  2d,     \ 

Parallelogram,  one  side  given,  by  prob.  4th,    f          . 
Triangle  of  a  given  base,     .     by  prob.  7th,    f 

Circle by  prob.  13th,  ) 

It  is  sometimes  most  convenient,  when  land  is  to  be  laid  out 
adjacent  to  a  creek,  river,  or  other  crooked  boundary,  to 
measure  offsets  to  the  angles  or  bending  thereof,  from  a  right 
line  or  lines  taken  near  such  boundary,  and  to  deduct  the  area 
of  these  offsets  from  the  given  quantity,  and  then  to  lay  off  the 
remainder  from  the  right  line  or  lines,  in  the  desired  form. 

In  laying  out  new  lands,  attention  must  be  paid  to  the  allow- 
ance for  roads,  as  exemplified  in  prob.  14th. 


SECTION  VIII. 
OF  SURVEYING  HARBOURS,  SHOALS,  SANDS,  &c. 

PL.  19.  fig.  1. 

THERE  are  three  methods  whereby  this  may  be  performed ; 
for  the  observations  may  be  made  either  on  the  water  or  on  the 
land.  Those  made  on  the  water  are  of  two  kinds  ;  one  by  the 
log-line  and  compass  (as  in  plane  sailing  measuring)  the  course 
and  distance  round  the  sand ;  and  then  to  be  plotted  as  a  large 
wood,  or  any  enclosure  taken  by  the  circumferentor. 

This  method  I  omit,  for  two  reasons  :  first,  because  it  is  to 
be  deduced  from  the  writers  of  navigation  ;  and,  secondly,  be- 
cause the  distances  thus  measured  are  liable  to  the  errors  of 
currents,  which  generally  attend  shoals  or  sands  near  the  shore. 
.  The  second  method,  when  there  are  no  distances  to  be 
measured  on  the  water,  though  still  there  is  one  inconvenience, 
common  also  to  the  former,  because  the  bearings  or  observa- 
tions are  to  be  taken  on  that  unstable  element  (an  error  scarce 
mentioned  by  practical  artists),  I  shall  briefly  hint  at ;  and  so 
rather  choose  a  third,  which  is  liable  to  neither  of  these  imper- 
fections. 

Let  a  boat  be  manned  out  with  a  signal  flag,  a  log  and  line, 
lead  and  line,  and,  to  observe  the  bearings  of  any  landmark,  a 
compass  with  sights. 

Take  two  or  more  objects  or  places,  as  A,  B,  C,  on  the 


OF  SURVEYING  HARBOURS,  <fcc.  221 

shore,  from  whence  the  boat  may  be  seen  on  the  several  parts 
of  this  shoal,  and  determine  their  relative  position  by  bearing 
and  distances  either  before  or  after  the  other  necessary  obser- 
vations are  made. 

One  of  the  boat's  crew  is  to  sound  till  he  finds  himself  on 
the  edge  of  the  sand,  by  the  depth  of  water,  and  then  to  come 
to  an  anchor ;  which  he  is  to  signify  to  two  persons  on  the 
shore  at  B  and  C,  by  his  signal.  And  then  from  those  known 
landmarks  B  and  C  the  observers  are  to  take  the  bearings  of 
the  boat,  and  to  register  their  observations  ;  which,  when  done, 
they  are  to  signify  to  the  crew  by  waving  a  flag,  or  by  some 
other  signal. 

And  in  the  mean  time,  to  prevent  mistakes,  let  the  crew  take 
the  bearings  of  each  of  these  landmarks :  then  weigh  anchor, 
which  suppose  at  D. 

Then,  by  sounding,  proceed  to  .E,  and  make  like  observa- 
tions. And  so  at  E,  F,  G,  &c.,till  you  have  surrounded  your 
sand. 

And  if  in  the  process  you  are  about  to  lose  the  sight  of  one 
of  your  landmarks,  suppose  C,  let  your  assistant  at  C,  or  B, 
who  at  that  time  will  also  be  about  to  lose  the  sight  of  the  boat, 
by  signals  (before  agreed  on)  remove  to  some  other  object 
beforehand  agreed  on,  suppose  to  H  or  K,  and  then  to  proceed 
as  before. 

Lastly,  if  the  sand  runs  so  for  out  at  sea  that  the  object  can- 
not be  seen  by  the  boat,  nor  the  boat  by  the  observer  on  shore, 
there  may  be  rockets  fired  by  the  boat's  crew,  and  also  by  the 
observers  on  the  shore  in  the  night,  whereby  those  bearings 
may  be  taken  almost  at  as  great  a  distance  as  the  light  can  be 
seen.  For  supposing  they  rise  but  a  quarter  of  a  mile  above 
the  apparent  horizon,  its  stay  will  be  about  9  seconds,  and  its 
distance  for  this  quarter  of  a  mile  will  be  visible  about  44  miles. 

But  rockets  rise  much  higher,  and  then  the  distances  are 
much  greater  whereby  they  are  visible. 

Or  two  boats  may  lay  at  anchor,  instead  of  the  landmarks, 
and  then  you  may  work  as  before. 

Now  since  the  landmarks  B  and  C  are  fixed,  their  position 
may  be  laid  down  in  the  draught,  as  in  common  surveyihg,  by 
plotting  the  distance  between  B  and  C.  And  then  by  plotting 
the  line  BD,  and  the  line  DC.  according  to  their  position,  their 
common  intersection  will  give  the  point  D.  And  in  like  man- 
ner £,  jP,  G,  &c.  may  be  plotted ;  and  so  the  shoals  completed. 
And  this  from  the  bearings  taken  at  B  and  C. 

If  this  be  a  standing  lake,  environed  by  bogs,  or  other  im- 
pediments, the  observations  at  D,  12,  JF1,  &c.,  by  taking  their 


222  OF  LEVELLING. 

opposites,  may  suffice  to  plot  the  same  from  the  landmarks,  A, 
B,  C,  &c.  as  well  as  those  taken  on  the  land :  or,  indeed,  by 
the  course  and  distance,  as  in  navigation,  if  the  water  be  smooth 
and  without  a  current. 

In  sea  shoals,  it  is  convenient  to  note  at  each  observation 
the  depth  of  the  water  found  by  the  lead,  and  the  drift  and 
setting  of  the  current  by  the  log  and  compass,  while  the  boat 
is  at  anchor,  which  may  be  done  with  ease  and  expedition 
enough.  For  while  the  boat  rides  at  an  anchor,  her  stern  points 
out  the  setting  of  the  current,  and  the  log  and  glass  will 
measure  its  drift. 

And  these  ought  to  be  noted  on  the  draught,  which  may  be 
thus: 

The  currents  may  be  shown,  by  drawing  a  dart  pointing 
out  its  setting,  and  its  drift  by  the  Roman  capital  letters,  the 
depth  of  the  water  by  the  small  figures,  and  rocks  by  little 
crosses,  &c. 


SECTION  IX. 
OF  LEVELLING. 

PL.  13.  Jig.  2. 

LEVELLING  is  the  art  of  ascertaining  the  perpendicular  ascent 
or  descent  of  one  place  (or  more)  above  or  below  the  horizon- 
tal level  of  another,  for  various  intentions,  and  of  marking  out 
courses  for  conveyance  of  water,  &c. 

The  true  level  is  a  curve  conforming  to  the  surface  of  the 
earth ;  as  ABG. 

The  apparent  level  is  a  tangent  to  that  curve  ;  as  ADE. 

The  correction  or  allowance  for  the  earth's  curvature  is  the 
difference  between  the  apparent  level  and  the  true ;  as  BD. 
The  quantity  of  this  correction  may  be  known  by  having  in  the 
right-angled  triangle  CAD  the  two  legs  AC  =  the  semidiameter 
of  the  earth  (  =  1267500  perches),  and  AD  =  the  distance  of  the 
object,  to  find  the  hypothenuse  CD,  from  which  taking  CB 
(  =  CA),  the  remainder  will  be  the  correction  BD\  but  it  may 
be  obtained  more  practically  thus  : 

~  ,     (  four-pole  chains,  and  divide  by  800,  }  for  the  cor- 

16  I  or  in  perches,  and  divide  by  12800  V  rection    in 
distance  in  J  ^  -n  ^^  &nd  multipiy  by  8  }  inches.    % 


OF  LEVELLING. 


223 


EXAMPLE. 

Required  the  correction  for  20  four-pole  chains  =  80  perches 
=  }  mile. 

800)20  X20=400(.5 
]  2800)80  X80=6400(.5 
| =.25,  and  .25  X  .25  X  8 =.5. 
That  is,  .5,  or  i  inch,  the  correction  required. 
But,  to  save  the  trouble  of  calculation,  we  insert  the  follow- 
ing table  of  corrections. 

A.  Table  of  Corrections. 
The  distances  in  four-pole  chains. 


Distan. 

Correc. 

Distan. 

Correc. 

Chains. 

Inches. 

Chains. 

Inches. 

1 

0.00125 

27 

0.91 

2 

0.005 

28 

0.98 

3 

0.01125 

29 

1.05 

4 

0.02 

30 

1.12 

5 

0.03 

31 

1.19 

6 

0.04 

32 

.27 

T 

0.06 

33 

1.35 

8 

0.08 

34 

.44 

9 

0.10 

35 

.53 

10 

0.12 

36 

.62 

11 

0.15 

37 

.71 

12 

0.18 

38 

.80 

13 

0.21 

39 

.91 

14 

0.24 

40 

2.00 

15 

0.28 

45 

2.28 

16 

0.32 

50 

3.12 

17 

0.36 

55 

3.78 

18 

0.40 

60 

4.50 

19 

0.45 

65 

5.31 

20 

0.50 

70 

6.12 

21 

0.55 

75 

7.30 

22 

0.60 

80 

8.00 

23 

0.67 

85 

9.03 

24 

0.72 

90 

10.12 

25 

0.78 

95 

11.28  . 

26 

0.84 

100 

12.50 

• 


224  OF  LEVELLING. 

The  first  thing  necessary  in  levelling  is  the  adjusting  of  the 
level,  which  may  be  performed  several  ways.  The  following 
is  very  easy  and  practical. 

Choose  some  ground  which  is  not  above  4  or  5  feet  out  of 
the  level,  for  the  distance  of  8  or  10  chains  in  length,  and 
suppose  it  be  AB  (fig.  3),  and  find  the  middle  between  A 
and  B,  which  suppose  to  be  C ;  plant  the  intstrument  at  C, 
direct  the  tube  to  a  station-staff  held  up  at  A,  and  elevate  or 
depress  the  tube  till  the  bubble  is  exactly  in  the  middle  of  the 
divisions ;  then  by  signals  direct  your  assistant  at  A  to  raise  or 
depress  the  vane  sliding  on  the  station-staff  till  the  horizontal 
hair  in  the  glass  cuts  the  middle  of  that  vane ;  then  see  how 
many  feet,  inches,  and  parts  are  cut  by  the  upper  part  of  the 
vane,  which  suppose  to  be  3  feet  4  inches  and  6  tenths. 

In  like  manner  direct  to  the  other  staff  at  B,  and  suppose 
the  upper  edge  of  that  vane  to  be  cut  at  the  height  of  6  feet  5 
inches  and  2  tenths,  then  will  these  two  vanes  be  on  a  level. 

From  6  feet  5.2  inches  subtract  3  Teet  4.6  inches,  and  re- 
serve the  remainder  3  feet  0.6  inches. 

Now  remove  the  instrument  as  close  to  the  higher  station- 
staff  as  you  can ;  so  that  the  middle  of  the  telescope  may 
almost  touch  it.  Then  bring  the  telescope  as  near  to  a  level 
as  the  judgment  of  the  eye  will  direct. 

Measure  from  the  ground  the  height  of  the  top  of  the  tele- 
scope ;  and  also  of  the  bottom  in  feet,  inches,,and  parts  ;  suppose 
them  to  be  4  feet  10.5  inches,  and  5  feet  0.3  inches ;  then  half 
€the  sum  of  the  heights  4  feet  1 1.4  inches  is  the  height  of  the  cen- 
tre of  the  glass ;  and  to  this  add  half  the  breadth  of  the  vane, 
which  suppose  to  be  1  inch  and  5  tenths,  and  to  the  sum  5  feet 
0.9  inches  add  the  preceding  remainder  3  feet  0.6  inches  ;  then 
let  the  person  at  B  move  his  vane  till  the  upper  edge  cut  8  feet 
1.5  inches,  the  sum  of  the  preceding  numbers. 

Now  so  elevate  or  depress  the  hair  of  the  bubble  till  the 
hair  cut  the  middle  of  the  vane  at  U,  and  at  the  same  time 
the  bubble  stands  at  the  middle  of  the  divisions  ;  and  then  will 
the  instrument  be  duly  adjusted. 

If  you  have  a  mind  to  be  more  accurate,  repeat  the  opera- 
tion ;  but  when  you  place  the  instrument  at  C,  turn  the  tube 
at  right  angles  to  the  lineal?,  and  there  set  it  level;  then 
proceed  with  a  repetition  of  the  work.  Only  observe  to  cross- 
level  it  in  this  adjustment,  and  in  all  future  uses  whatsoever. 

Or  the  level  maybe  adjusted  thus  :  As  before,  first  plant  the 
instrument  in  the  middle  between  A  and  B  (fig.  4),  and  ob- 
serve the  heights  on  the  station-staves,  which  suppose  to  be  as 
abQve ;  and  consequently  their  difference,  as  before,  is  3  feet 
0.6  inches.  Now  measure  from  C  towards  the  highest  ground 


OF  LEVELLING.  225 

A,  some  distance  that  comes  almost  to  A,  suppose  4  chains  to 
D ;  and  DB  will  be  9  chains,  and  DA  one  chain ;  then  plant  the 
instrument  at  Z),  direct  the  telescope  to  A,  and  setting  the 
bubble  to  the  middle  of  the  division  direct  your  assistant  to 
move  the  vane  till  the  hair  cuts  the  middle  of  it,  and  note  down 
the  feet,  inches,  and  parts  cut  by  the  upper  edge  of  the  vane, 
which  suppose  to  be  3  feet  8.4  inches  :  to  this  add  the  differ- 
ence 3  feet  0.6  inches,  and  the  sum  6  feet  9  inches  reserve. 

Now  direct  the  telescope  to  the  staff  at  .B,  level  it,  and 
direct  your  assistant  to  move  the  vane  till  the  hair  cuts  the 
middle  thereof;  and  then  if  the  upper  edge  of  the  vane  cuts 
the  foregoing  siim  6  feet  9  inches,  the  hair  and  bubble  are  truly 
adjusted.  But  if  not,  say,  as  BD  less  AD  is  to  the  difference 
between  the  numbers  cut  by  the  upper  edge  of  the  vane  and 
the  number  6  feet  9  inches,  so  is  the  distance  AD  to  a  number 
which,  added  to  that  cut  by  the  vane  when  less  than  6  feet  9, 
and  subtracted  from  the  number  cut  by  the  vane  when  it  is  greater 
than  6  feet  9,  will  give  a  number,  to  which  let  the  assistant  fix  the 
vane  ;  then  so  elevate  or  depress  the  hair  or  the  bubble  till  the 
hair  cuts  the  middle  of  the  vane  at  B,  and  the  bubble  stands  in 
the  middle  of  the  divisions ;  for  then  the  level  will  be  adjusted. 
The  operation  may  be  again  repeated,  and  at  every  station 
cross-levelled,  which  will  confirm  the  former  adjustment. 

Or  it  will  be  still  better  to  set  the  station-staves  equally  dis* 
tant  from  the  instrument  (suppose  about  16  or  20  perches  each) 
at  an  angle  of  about  60°,  or  so  as  to  form  nearly  an  equilateral 
triangle  therewith,  and  level  the  two  vanes  ( A  and  B,  fig.  5),  as 
before,  which  will  be  then  both  in  the  same  horizontal  level, 
whether  the  instrument  be  rightly  adjusted  or  not,  because  one 
will  be  as  much  above  or  below  the  true  level  of  the  instru- 
ment as  the  other,  being  in  the  same  distance  from  it ;  then 
remove  the  instrument  as  near  as  may  be  to  one  of  them,  sup- 
pose A,  and  raise  or  lower  the  vane  A  to  the  exact  level  of 
the  visual  ray  in  the  instrument,  noting  precisely  how  much  it 
is  moved,  and  have  the  other  vane  B  moved  just  as  much,  in 
order  to  bring  them  again  to  a  level,  allowing  for  the  correction 
of  the  apparent  level  if  it  be  a  sensible  quantity ;  then  adjust 
the  instrument  to  the  level  of  the  vane  at  B. 

To  adjust  the  rafter-level  (plate  13,  fig.  6),  which  maybe 
10,  12,  or  14  feet  in  the  span  AB ;  set  it  on  a  plank  or  hard 
ground  nearly  level,  and  mark  where  the  plumb  line  cuts  the 
beam  mn,  suppose  at  c ;  then  invert  the  position  by  setting 
the  foot  A  in  the  place  of  U,  and  B  in  that  of  j4,  marking 
where  the  line  now  cuts,  as  at  e ;  the  middle  point  between  c 
and  e  will  be  the  true  levelling  mark.  * 

K3 


226  OF  LEVELLING. 

To  continue  a  level  course  with  this  instrument,  set  the 
foot  A  to  the  starting  place,  and  move  B  upward  or  downward 
towards  D  or  E,  till  the  point  B  be  determined  and  marked 
for  a  level  with  A ;  then  carry  the  instrument,  forward  in  the 
direction  of  C,  till  the  foot  A  rests  at  2?,  whence  the  point  C 
is  levelled  as  before,  &c.  Sights  may  be  placed  at  r  and  s, 
and  the  instrument  adjusted  to  them,  as  before,  by  reversing 
them  in  the  direction  of  some  distant  object. 

After  the  instrument  is  duly  adjusted,  you  may  proceed  to 
use  it.  Let  the  example  be  this  annexed  (fig.  7),  where  A 
everywhere  represents  the  level,  and  B  the  station-staves  ;  and 
suppose  the  route  be  made  from  a  to  e  :  first  plant  the  instru- 
ment between  the  staves  a  and  b  ;  at  A  direct  the  level  to  aB, 
bring  the  bubble  to  the  middle  of  the  divisions,  and  instruct 
your  assistant  so  to  place  the  vane  that  the  hair  in  the  tele- 
scope cuts  the  middle  of  the  vane  ;  then  in  a  book  divided  into 
two  columns,  the  one  entitled  Back-sights,  the  other  Fore-sights, 
enter  the  feet,  inches,  and  parts  cut  by  the  upper  edge  of  the 
vane  at  aB  in  the  column  entitled  Back-sights. 

Then  look  towards  the  other  staff  bB,  bring  the  bubble  to 
the  middle  of  the  divisions,  and  direct  your  assistant  to  place 
the  vane  so  that  the  hair  cuts  the  middle  of  the  vane ;  then 
enter  the  feet,  inches,  and  parts  cut  by  the  upper  edge  of  the 
vane  in  the- column  of  Fore-sights. 

Now  plant  the  instrument  at  A2,  still  keeping  the  staff  Bb 
exactly  in  the  same  place,  and  carry  the  staff  aB  forward  to 
the  place  cB ;  now  look  back  to  the  staff  bB,  and  enter  the 
numbers  cut  by  the  vane  there  under  the  title  Back-sights ; 
then  look  forwards  to  cB,  and  enter  the  observation  under 
the  title  Fore-sights.  Do  the  like  when  the  instrument  is 
planted  at  A*,  A4,  <fcc.,  always  taking  care  to  keep  the  staff  in 
the  same  place  when  you  looked  at  it  for  a  fore-sight,  till  you 
have  also  taken  with  it  a  back-sight. 

Having  finished  your  level,  add  up  the  column  of  back' 
sights  into  one  sum,  and  the  column  of  fore-sights  also  into 
one  sum ;  and  the  difference  between  these  sums  is  the  ascent 
or  descent  required.  And  if  the  sum  of  the  fore-sights  be 
greater  than  the  sum  of  the  back-sights,  then  e  is  lower  than 
a ;  but  if  the  sum  of  the  fore-sights  be  less  than  the  sum  of 
the  back-sights,  e  is  higher  than  a.  For  example,  let  the 
numbers  be  as  in  the  following  table. 


OF  LEVELLING. 


Back-sights. 

Fore-sights. 

Feet.  Inches.  Tenths. 

Feet. 

Inches 

.Tenths. 

3.7      .5 

6 

.     4 

.      5 

4.6.8 

8 

.     3 

.      2 

6.0.2 

5 

.     4 

.      7 

9.5.0 

8 

.     7 

.      8 

1.0.7 

9 

.     4 

.      8 

24     .     8      .      2 

38 

.     1 

.     0 

24 

.     8 

.     2 

Hence  the  descent  is  <    ' 

.     4 
.     4 

.     8 
.      8 

Observations. 

1.  And  if  the  distances  thus  taken  are  short,  the  curvature 
of  the  earth  may  be  rejected.     For,  if  the  distance  from  the 
instrument  be  everywhere  about  100  yards,  all  the  curvatures 
in  a  mile's  work  will  be  less  than  half  an  inch. 

2.  If  the  distance  from  the  instrument  to  the  hindermost 
staff  be  everywhere  equal  to  the  distance  from  the  instrument 
to  the  corresponding  staff,  the  curvature  of  the  earth  and  the 
minute  errors  of  the  instrument  will  both  be  destroyed.    Hence 
it  will  be  much  better  to  set  the  instrument  as  equally  distant 
from  both  staves  as  may  be. 

3.  If  the   distances  of  the  instrument  from  the  staves  be 
very  unequal  and  very  long,  the  curvatures  must  be  accounted 
for,  and  the  distances,  in  order  thereto,  must  be  measured. 

4.  Therefore  it  appears,  that  the  best  method  to  take  a  level 
is,  to  measure  the  several  distances  from  the  instrument  to  the 
back  and  forward  station-staves ;  and  enter  them  in  the  field- 
book,  according  to  the  titles  of  their  several  columns,  as  in  the 
following  example  ;  and  correct  the  heights  from  the  table  of, 
allowances,  which  may  be  done  at  home  when  you  are  about 
to  sum  up  the  heights 


228 


OF  LEVELLING. 


Backwards. 

Forwards. 

Distan. 

Height 

Corrected. 

Distan. 

Height. 

Corrected. 

Links. 

Inches. 

Inches. 

Links. 

Inches. 

Inches. 

370 

3.25 

3-24 

418 

4.36 

4.34 

420 

6.10 

6.08 

328 

7.18 

7.17 

760 

5.38 

5.31 

289 

6.75 

6.67 

584 

7.25 

7.21 

530 

9.53 

9.50 

326 

8.15 

8.14 

485 

11.25 

11.22 

658 

10.25 

10.20 

376 

8.65 

8.63 

530 

6.32 

6,29 

720 

10.34 

10.28 

36.58 

46.47 

31.46 

57.81 

31.46 

46.47 

68.04 

11.34 

So  that  the  fall  in  68  chains  is  about  1 1  inches  and  1  of  an 
inch. 

Lastly,  though  hitherto  we  have  considered  the  level  with 
one  telescope  only,  the  same'  observations  may  be  applied  to  a 
level  with  a  double  telescope ;  and  I  would  advise  those  who 
use  the  double  telescope,  at  every  station  to  turn  that  end  of 
the  telescope  forward  which  before  was  the  contrary  way. 


A  more  general  method,  of  levelling,  adapted  to  the  surveying  of  roads 
and  hilly  ground,  is  exhibited  in  the  following  example,  in  which  the  measures 
ure  given  in  links. 

EXAMPLE. 

PL.  IB.  fig.  S. 

Required  the  bearing  and  distance  of  the  place  B  from  A, 
and  its  perpendicular  ascent  or  descent  above  or  below  the 
horizontal  level  of  A. 


OF  LEVELLING. 


229 


St. 

Course  or 
Bearing. 

Elevation  or 
Depression. 

Obi. 
Dist. 

Hor. 
Dist. 

Pefpen. 
Ascent 
or  Des. 

Dif. 
Lat. 

0 
* 

1 

N.E.79°15' 

D.  17°  15' 

738 

705 

218.9 

131 

692 

2 

N.E.75  00 

D.  21    45 

684 

635 

253.4 

164 

613 

3 

N.E.  50  30 

E.  14   00 

976 

947 

236.1 

602 

730 

4 

S.E.  85   15 

D.  11    30 

930 

911 

185.4 

75 

908 

5 

S.E.  70  00 

E.  19    15 

620 

585 

204.0 

200 

549 

3948 

3783 

217.6 

622 

3492 

Desc. 

N. 

E. 

As  dif.  lat.  622 

Is  to  radius  S.  20°, 
So  is  dep.  3492 

To  T.  bear.  79°  54'. 


As  S.  bear.  79°  45' 
Is  to  dep.  3492, 

So  is  radius  S.  90° 
To  dist.  3547. 


As  100  links  :  66  feet  : 
below  the  level  of  A. 


217.6  links  :  143.6  feet,  the  descent 


Hence,  B  bears  N.  79°  54'  E.  from  A. 
Nearest  horiz.  dist.  3547  links. 
Sum  of  obi.  dist.  3948  links. 
Sum  of  horiz.  dist.  3783  links. 


>  Answer. 

Perp.  desc.  2 17.6  links= 143.6ft.  J 

With  the  angular  elevation  or  depression  in  the  third  column, 
and  the  oblique  distance  in  the  fourth  (as  course  and  distance) 
are  found  the  horizontal  distance  in  the  fifth,  and  the  perpen- 
dicular ascent  or  descent  on  the  sixth,  for  each  station  (as  dif- 
ference of  latitude  and  departure)  :  then,  with  the  bearing  and 
horizontal  distance,  we  get  the  difference  of  latitude  and  de- 
parture in  the  last  two  columns. 

The  ascents  and  descents  in  the  sixth  column  are  distin- 
guished by  the  letters  E  and  D  in  the  third,  signifying  eleva- 
tion or  depression ;  and  being  added  separately,  the  difference 
of  their  sums  is  set  at  the  bottom  of  the  column  with  the  name 
of  the  greater,  and  shows  the  perpendicular  descent  of  B  below 
the  horizontal  level  of  A. 

In  like  manner  the  northings  and  southings  in  the  seventh 
column  are  distinguished  by  the  letters  N  and  S  in  the 
second,  &c. 

PROMISCUOUS  QUESTIONS. 

1.  The  perambulator,  or  surveying  wheel,  is  so  contrived  as 


230  PROMISCUOUS  QUESTIONS. 

to  turn  just  twice  in  the  length  of  a  pole,  or  16£  feet;  what 
then  is  the  diameter  ?  Ans.  2.626  feet. 

2.  Two  sides  of  a  triangle  are  respectively  20    and  40 
perches ;  required  the  third,  so  that  the  contents  may  be  just  an 
acre.  Ans.  either  23.099  or  58.876  perches. 

3.  I  want  the  length  of  a  line  by  which  my  gardener  may 
strike  out  a  round  orangery  that  shall  contain  just  half  an  acre 
of  ground.  Ans.  27f  yards. 

4.  What  proportion  does  the  arpent  of  France,  which  con- 
tains 100  square  poles  of  18  feet  each,  bear  to  the  American 
acre,  containing  160  square  poles  of  16.5  feet  each,  considering 
that  the  length  of  the  French  foot  is  to  the  American  as  16 
to  15?  Ans.  as  512  to  605. 

5.  The  ellipse  in  Grosveiiur  Square  measures  840  links  the 
longest  way,  and  612  the  shortest,  within  the  rails :  now  the 
wall  being  14  inches  thick,  it  is  required  to  find  what  quantity 
of  ground  it  encloses,  and  how  much  it  stands  upon. 

Ans.  It  encloses  4 A.  6P.,  and  stands  on  1760^  square  feet. 

6.  Required  the  dimensions  of  an  elliptical  acre  with  the 
..greatest  and  least  diameters  in  the  proportion  of  3  to  2. 

Ans.  17.479  by  11.  653  perches. 

7.  The  paving  of  a  triangular  court  at  ISd.  per  foot,  came 
to   WOl.     The  longest  of  the  three  sides  was  88  feet:  what 
then  was  the  sum  of  the  other  two  equal  sides  1 

Ans.  106.85  feet. 

8.  In  110  acres  of  statute  measure,  in  which  the  pole  is  161 
feet,  how  many  Cheshire   acres,  where  the  customary  pole 
is  6  yards,  and  how  many  of  Ireland,  where  the  pole  in  use  is 
7  yards  ? 

Ans.  92A.  1R.  28P.  Cheshire ;  67A.  3R.  25  P.  Irish. 

9.  The  three  sides  of  a  triangle  containing  6A.  1R.  12P.  are 
in  the  ratio  of  the  three    numbers  9,  8,  6,  respectively ;  re- 
quired the  sides.  Ans.  59.029,  52.47,  and  39.353. 

10.  In  a  pentangular  field,  beginning  with  the  south  side, 
and  measuring  round  towards  the  east,  the  first  or  south  side  is 
2735  links,  the  second  3115,  the  third  2370,  the  fourth  2925* 
and  the  fifth  2220 ;  also  the  diagonal  from  the  first  angle  to 

•  the  third  is  3800  links,  and  that  from  the  third  to  the  fifth  4010 ; 
required  the  area  of  the  field.  Ans.  117A.  2R.  28P. 

11.  Required  the  dimensions  of  an  oblong  garden  containing 
three  acres,  and  bounded  by  104  perches  of  pale  fence. 

Ans.  40  perches  by  12. 

12.  How  many  acres  are  contained  in  a  square  meadow,  the 
diagonal  of  which  is  20  perches  more  than  either  of  its  sides  t 

Ans.  14A.  2R.  IIP. 


INTRODUCTORY  PRINCIPLES.  23t 

13.  If  a  man  six  feet  high  travel  round  the  earth,  how  much 
greater  will  be  the  circumference  described  by  the  top  of  his 
head  than  by  his  feet?  Ans.  37.69  feet. 

N.  B. — The  required  difference  is  equal  to  the  circumference 
of  a  circle  6  feet  radius,  let  the  magnitude  of  the  earth  be  what 
it  may. 

14.  Required  the  dimensions  of  a  parallelogram  containing 
200  acres,  which  is  40  perches  longer  than  wide. 

Ans.  200  perches  by  160. 

15.  What  difference  is  there  between  a  lot  28  perches  long 
by  20  broad,  and  two  others,  each  of  half  the  dimensions  ? 

Ans.  1A.  3R. 


PART   III. 

Containing  the  astronomical  methods  of  finding  the  latitude,  variation 
of  the  compass,  $c.,  with  a  description  of  the  instruments  used  in  these 
operations. 

SECTION  I. 

INTRODUCTORY  PRINCIPLES. 

DAY  and  night  arise  from  the  circumrotation  of  the  earth. 
That  imaginary  line  about  which  the  rotation  is  performed  is 
called  the  axis,  and  its  extremities  are  called  poles.  That 
towards  the  most  remote  parts  of  Europe  is  called  the  north 
pole,  and  its  opposite  the  south  pole.  The  earth's  axis  being 
produced  will  point  out  the  celestial  poles. 

The  equator  is  a  great  circle  on  the  earth,  every  point  of 
which  is  equally  distant  from  the  poles ;  it  divides  the  earth 
into  two  equal  parts,  called  hemispheres  :  that  having  the  north 
pole  in  its  centre  is  called  the  northern  hemisphere,  and  the 
other  the  southern  hemisphere.  The  plane  of  this  circle  being 
produced  to  the  fixed  stars  will  point  out  the  celestial  equator, 
or  equinoctial.  The  equator,  as  well  as  all  other  great  circles  of 
the  sphere,  is  divided  into  360  equal  parts,  called  degrees ;  each 
degree  is  divided  into  60  equal  parts,  called  minutes  ;  and  the 
sexagesimal  division  is  continued. 

Note.^- The  ancients,  having  no  instruments  by  which  they 
could  make  observations  with  any  tolerable  degree  of  accuracyt 
supposed  the  length  of  the  year,  or  annual  motion  of  the  earth,, 
to  be  completed  in  360  days :  and  hence  arose  the  division  of 


232  INTRODUCTORY  PRINCIPLES. 

the  circumference  of  a  circle  into  the  same  number  of  equal[ 
parts,  which  they  called  degrees. 

The  meridian  of  any  place  is  a  semicircle  passing  througn1' 
that  place,  and  terminating  at  the  poles  of  the  equator.  The 
other  half  of  this  circle  is  called  the  opposite  meridian. 

The  latitude  of  any  place  is  that  portion  of  the  meridian  of 
that  place  which  is  contained  between  the  equator  and  the 
given  place ;  and  is  either  south  or  north,  according  as  the 
given  place  is  in  the  northern  or  southern  hemisphere,  and  there- 
fore cannot  exceed  90°. 

The  parallel  of  latitude  of  any  place  is  a  circle  passing 
through  that  place  parallel  to  the  equator.  ( 

The  difference  of  latitude  between  any  two  places  is  an 
arch  of  a  meridian  intercepted  between  the  corresponding  paral- 
lels of  latitude  of  those  places.  Hence,  if  the  places  lie  be- 
tween the  equator  and  the  same  pole,  their  difference  of  lati- 
tude is  found  by  subtracting  the  less  latitude  from  the  greater ; 
but  if  they  are  on  opposite  sides  of  the  equator,  the  difference 
of  latitude  is  equal  to  the  sum  of  the  latitudes  of  both  places. 

The  first  meridian  is  an  imaginary  semicircle,  passing 
through  any  remarkable  place,  and  is  therefore  arbitrary. 
Thus,  the  British  esteem  that  to  be  the  first  meridian  which  passes 
through  the  royal  observatory  at  Greenwich  ;  and  the  French 
reckon  for  their  first  meridian  that  which  passes  through  the 
royal  observatory  at  Paris. — Formerly  many  French  geogra- 
phers reckoned  the  meridian  of  the  island  of  Ferro  to  be  their 
first  meridian;  and  others,  that  which  was  exactly  20  de- 
grees to  the  west  of  the  Paris  observatory.  The  Germans, 
again,  considered  the  meridian  of  the  Peak  of  Teneriffe  to  be 
the  first  meridian.  By  this  mode  of  reckoning,  Europe,  Asia, 
and  Africa  are  in  east  longitude,  and  North  and  South  America 
in  west  longitude.  At  present  the  first  meridian  of  any  coun- 
try is  generally  esteemed  to  be  that  which  passes  through  the 
principal  observatory,  or  chief  city,  of  that  country.  i 

The  longitude  of  any  place  is  that  portion  of  the  equator 
which  is  contained  between  the  first  meridian  and  the  meridian 
of  that  place ;  and  is  usually  reckoned  either  east  or  west,  ac- 
cording as  the  given  place  is  on  the  east  or  west  side  of  the 
first  meridian;  and,  therefore,  cannot  exceed  180°. 

The  difference  of  longitude  between  any  two  places  is  the 
intercepted  arch  of  the  equator  between  the  meridians  of  those 
places,  and  cannot  exceed  180°. 

There  are  three  different  horizons,  the  apparent,  the  sensi- 
ble, and  the  true.  The  apparent  or  visible  horizon  is  the  ut^ 
most  Apparent  view  of  the  sea  or  land}  the  sensible  is  a  plane 


INTRODUCTORY  PRINCIPLES.  233 

passing  through  the  eye  of  an  observer,  perpendicular  to  a 

plumb-line  hanging  freely ;  and  the  true  or  rational  horizon  is 

a  plane  passing  through  the  centre  of  the  earth,  parallel  to  the 

sensible  horizon. 

;     Altitudes  observed  at  sea  are   measured  from  the  visible 

horizon.     At  land,  when  an  astronomical  quadrant   is  used,  or 

when  observations  are  taken  with  a  Hadley's  quadrant  by  the 

method  of  reflection,  the  altitude  is  measured  from  the  sensible 

horizon  ;  and  in  either  case  the  altitude  must  be  reduced  to  the 

true  horizon. 

i     The   zenith  of  any  given  place  is  the  point  immediately 

above  that  place,  and  is,  therefore,  the  elevated  pole  of  the 

horizon.     The  nadir  is  the  other  pole,  or  point  diametrically 

opposite. 

i     A  vertical  is  a  great  circle  passing  through  the  zenith  and 

nadir  ;  and  therefore  intersecting  the  horizon  at  right  angles. 

The  altitude  of  any  celestial  body  is  that  portion  of  a  ver- 
tical which  is  contained  between  its  centre  and  the  true  hori- 
zon. The  meridian  altitude  is  the  distance  of  the  object  from 
the  true  horizon,  when  on  the  meridian  of  the  place  of  obser- 
vation. When  the  observed  altitude  is  corrected  for  the  de- 
pression of  the  horizon  and  the  errors  arising  from  the  instru- 
ment, it  is  called  the  apparent  altitude ;  and  when  reduced  to 
the  true  horizon,  by  applying  the  parallax  in  altitude,  it  is 
called  the  true  altitude.  Altitudes  are  expressed  hi  degrees  and 
parts  of  a  degree. 

The  zenith  distance  of  any  object  is  its  distance  from  the 
zenith,  or  the  complement  of  its  altitude. 
,  The  decimation  of  any  object  is  that  portion  of  its  meridian 
which  is  contained  between  the  equinoctial  and  the  centre  of 
the  object ;  and  is  either  north  or  south  according  as  the  star 
is  between  the  equinoctial  and  the  north  or  south  pole. 

The  ecliptic  is  that  great  circle  in  which  the  annual  revolu- 
tion of  the  earth  round  the  sun  is  performed.  It  is  so  named 
because  eclipses  cannot  happen  but  when  the  moon  is  in  or  near  .. 

that  circle.  The  inclination  of  the  ecliptic  and  equinoctial  is 
at  present  about  23°  28' ;  and  by  comparing  ancient  with  mod- 
ern observations,  the  obliquity  of  the  ecliptic  is  found  to  be 
diminishing — which  diminution,  in  the  present  century,  is  about 
half  a  second  yearly. 

The  ecliptic,  like  all  other  great  circles  of  the  sphere,  is  di- 
vided into  360°  ;  and  is  further  divided  into  twelve  equal  parts, 
called  signs :  each  sign,  therefore,  contains  30°.  The  names 
and  characters  of  these  signs  are  as  follows  ;  • 


234  INTRODUCTORY  PRINCIPLES. 


Aries, 

Taurus, 

Gemini, 


Cancer,  ss 
Leo,  £i 
Virgo,  UK 


Libra,  ess 

Scorpio,        TT], 
Sagittarius,  £ 


Capricornus, 

Aquarius, 

Pisces, 


Since  the  ecliptic  and  equinoctial  are  great  circles,  they 
therefore  bisect  each  other  in  two  points,  which  are  called  the 
equinoctial  points.  The  sun  is  in  one  of  these  points  in  March, 
and  in  the  other  in  September ;  hence,  the  first  is  called  the 
vernal,  and  the  other  the  autumnal  equinox — and  that  sign 
which  begins  at  the  vernal  equinox  is  called  Aries.  Those 
points  of  the  ecliptic  which  are  equidistant  from  the  equinoc- 
tial points  are  called  the  solstitial  points ;  the  first  the  summer, 
and  the  second  the  winter  solstice.  That  great  circle  which 
passes  through  the  equinoctial  points  and  the  poles  of  the  earth 
is  called  the  equinoctial  colure ;  and  the  great  circle  which 
passes  through  the  solstitial  points  and  the  poles  of  the  earth 
is  called  the  solstitial  colure. 

When  the  sun  enters  Aries  it  is  in  the  equinoctial,  and 
therefore  has  no  declination.  From  thence  it  moves  forward 
in  the  ecliptic,  according  to  the  order  of  the  signs,  and  ad- 
vances towards  the  north  pole,  by  a  kind  of  retarded  motion, 
till  it  enters  Cancer,  and  is  then  most  distant  from  the  equinoc- 
tial ;  and  moving  forward  in  the  ecliptic,  the  sun  apparently 
recedes  from  the  north  pole  with  an  accelerated  motion  till  it 
enters  Libra,  and,  being  again  in  the 'equinoctial,  has  no  decli- 
nation ;  the  sun,  moving  through  the  signs  Libra,  Scorpio,  and 
Sagittarius,  enters  Capricorn ;  and  then  its  south  declination  is 
greatest,  and  is,  therefore  most  distant  from  the  north  pole ; 
and  moving  forward  through  the  signs  Capricorn,  Aquarius,  and 
Pisces,  agajn  enters  Aries :  hence  a  period  of  the  seasons  is 
completed,  and  this  period  is  called  a  solar  year. 

The  signs  Aries,  Taurus,  Gemini,  Cancer,  Leo,  and  Virgo 
are  called  northern  signs,  because  they  are  contained  in  that 
part  of  the  ecliptic  which  is  between  the  equinoctial  and  north 
pole  ;  and,  therefore,  while  the  sun  is  in  these  signs,  its  decli- 
nation is  north :  the  other  six  signs  are  called  southern  signs. 
The  signs  in  the  first  and  fourth  quarters  of  the  ecliptic  are 
called  ascending  signs,  because  while  the  sun  is  in  these  signs 
it  approaches  the  north  pole ;  arid,  therefore,  in  the  northern, 
temperate,  and  frigid  zones,  the  sun's  meridian  altitude  daily 
increases ;  or,  which  is  the  same,  the  sun  ascends  to  a  greater 
height  above  the  horizon  every  day.  The  signs  in  the  second 
and  third  quarters  of  the  ecliptic  are  called  descending  signs. 

The  tropics  are  circles  parallel  to  the  equinoctial,  whos.e 
distance  therefrom  is  equal  to  the  obliquity  of  the  ecliptic* 


THE  QUADKAJNT.  23S 

The  northern  tropic  touches  the  ecliptic  at  the  beginning  of 
Cancer,  and  is  therefore  called  the  tropic  of  Cancer ;  and  the 
southern  tropic  touches  the  ecliptic  at  the  beginning  of  Capri- 
corn, and  is  hence  called  the  tropic  of  Capricorn. 

Circles  about  the  poles  of  the  equinoctial,  and  passing 
through  the  poles  of  the  ecliptic,  are  called  polar  circles  ;  the 
distance,  therefore,  of  each  polar  circle  from  its  respective  pole 
is  equal  to  the  inclination  of  the  ecliptic  and  equinoctial.  That 
circle  which  circumscribes  the  north  pole  is  called  the  arctic 
or  north  polar  circle ;  and  that  towards  the  south  pole,  the  ant- 
arctic or  south  polar  circle. 

That  semicircle  which  passes  through  a  star,  or  any  given 
point  of  the  heavens,  and  the  poles  of  the  ecliptic,  is  called  a 
circle  of  latitude. 

The  reduced  place  of  a  star  is  that  point  of  the  ecliptic 
which  is  intersected  by  the  circle  of  latitude  passing  through 
that  star. 

The  latitude  of  a  star  is  that  portion  of  the  circle  of  latitude 
contained  between  the  star  and  its  reduced  place  ;  and  is  either 
north  or  south,  according  as  the  star  is  between  the  ecliptic  and 
the  north  or  south  pole  thereof. 

The  longitude  of  a  star  is  that  portion  of  the  ecliptic  con- 
tained between  the  vernal  equinox  and  the  reduced  place  of 
the  star. 


SECTION  II. 
Description  of  the  instruments  requisite  in  astronomical  observations 

THE  QUADRANT. 

IT  is  generally  allowed  that  we  are  indebted  to  John  Hadley, 
Esq.  for  the  invention,  or  at  least  for  the  first  public  account, 
of  that  admirable  instrument  commonly  called  Hadley's  quad- 
rant, who  in  the  year  1731  first  communicated  its  principles  to 
the  Royal  Society,  which  were  by  them  published  soon  after 
in  their  Philosophical  Transactions ;  before  this  period  the 
cross-staff  and  Davis's  quadrant  were  the  only  instruments 
used  for  measuring  altitudes  at  sea,  both  very  imperfect,  and 
liable  to  considerable  error  in  rough  weather ;  the  superior  ex- 
cellence, however,  of  Hadley's  quadrant  soon  obtained  its. 


236  THE  QUADRANT. 

general  use  among  seamen,  and  the  many  improvements  this 
instrument  has  received  from  ingenious  men  at  various  times 
have  rendered  it  so  correct,  that  it  is  now  applied,  with  the 
greatest  success,  to  the  important  purposes  of  ascertaining  both 
the  latitude  and  longitude  at  sea  or  land. 

Figure  2,  Frontispiece,  represents  a  quadrant  of  reflection, 
the  principal  parts  of  which  are,  the  octant  or  frame  ABC 
(which  is  generally  made  of  ebony,  or  other  hard  wood,  and 
consists  of  an  arch  firmly  attached  to  two  radii  or  bars,  which 
are  strengthened  and  bound  by  the  two  braces  in  order  to  pre- 
vent it  from  warping),  the  graduated  arch  jBC,  the  index  D,  the 
nonius  or  vernier  scale  £,  the  index  glass  jP,  the  horizon  glasses 
G  and  H,  the  dark  glasses  or  screens  /,  and  the  sight  vanes 
K  and  L. 

The  arch,  or  limb  J5C,  although  only  the  eighth  part  of  a 
circle,  is,  on  account  of  the  double  reflection,  divided  into  90 
degrees,  numbered  0,  10,  20,  30,  &c.,  from  the  right  towards 
the  left :  these  are  subdivided  into  three  parts,  containing  each 
20  minutes,  which  are  again  subdivided  into  single  minutes,  by 
means  of  a  scale  at  the  end  of  the  index.  The  arch  extending 
from  0  towards  the  right-hand  is  called  the  arch  of  excess. 

The  index  D  is  a  flat  brass  bar,  that  turns  on  the  centre  of 
the  instrument ;  at  the  lower  end  of  the  index  there  is  an  ob- 
long opening ;  to  one  side  of  this  opening  a  nonius  scale  is 
fixed,  to  subdivide  the  divisions  of  the  arch ;  at  the  bottom  or 
end  of  the  index  there  is  a  piece  of  brass  which  bends  under 
the  arch,  carrying  a  spring  to  make  the  nonius  scale  lie  close 
to  the  divisions ;  it  is  also  furnished  with  a  screw  to  fix  the 
index  in  any  desired  position. 

Some  instruments  have  an  adjusting  or  tangent-screw,  fitted 
to  the  index,  that  it  may  be  moved  more  slowly,  and  with 
greater  regularity  and  accuracy  than  by  the  hand  ;  it  is  proper, 
however,  to  observe,  that  the  index  must  be  previously  fixed 
near  its  right  position  by  the  above-mentioned  screw,  before  the 
adjusting  screw  is  put  in  motion. 

The  nonius  is  a  scale  fixed  to  the  end  of  the  index,  for  the 
purpose,  as  before  observed,  of  dividing  the  subdivisions  on  the 
arch  into  minutes  ;  it  sometimes  contains  a  space  of  7  degrees, 
or  21  subdivisions  of  the  limb,  and  is  divided  into  20  equal 
parts  ;  hence*each  division  on  the  nonius  will  be  one-twentieth 
part  greater,  that  is,  one  minute  longer,  than  the  divisions  on 
the  arch ;  consequently,  if  the  first  division  of  the  nonius, 
marked  0,  be  set  precisely  opposite  to  any  degree,  the  relative 
position  of  the  nonius  and  the  arch  must  be  altered  one 
minute,  before  the  next  division  on  the  nonius  will  coincide 


THE  QUADRANT.  237 

with  the  next  division  on  the  arch,  the  second  division  will 
require  a  change  of  two  minutes,  the  third  of  three  minutes,  and 
so  on,  till  the  20th  stroke  on  the  nonius  arrives  at  the  next  20 
minutes  on  the  arch  ;  the  0  on  the  nonius  will  then  have  moved 
exactly  20  minutes  from  the  division  whence  it  set  out,  and  the 
intermediate  divisions  of  each  minute  have  been  regularly 
pointed  out  by  the  divisions  of  the  nonius. 

The  divisions  of  the  nonius  scale  are  in  the  above  case 
reckoned  from  the  midflle  towards  the  right,  and  frorh  the  left 
towards  the  middle;  therefore  the  first  10  minutes  are  con- 
tained on  the  right  of  the  0,  and  the  other  10  on  the  left.  But 
this  method  of  reckoning  the  divisions  being  found  inconvenient, 
they  are  more  generally  counted  beginning  from  the  right- 
hand  towards  the  left ;  and  then  20  divisions  on  the  nonius 
are  equal  to  19  on  the  limb,  consequently  one  division  on  the 
arch  will  exceed  one  on  the  nonius  by  one-twentieth  part,  that 
is,  one  minute. 

The  0  on  the  nonius  points  out  the  entire  degrees  and  odd 
twenty  minutes  subtended  by  the  objects  observed ;  and  if  it 
coincides  with  a  division  on  the  arch,  points  out  the  required 
angle  :  thus,  suppose  the  0  on  the  nonius  stands  at  25  degrees, 
then  25  degrees  will  be  the  measure  of  the  angles  observed ; 
if  it  coincides  with  the  next  division  on  the  left-hand,  25  de- 
grees 20  minutes  is  the  angle ;  if  with  the  second  division 
beyond  25  degrees,  then  the  angle  will  be  25  degrees  40 
minutes ;  and  so  on  in  every  instance  where  the  0  on  the  no- 
nius coincides  with  a  division  on  the  arch ;  but  if  it  does  not 
coincide,  then  look  for  a  division  on  the  nonius  that  stands 
directly  opposite  to  one  on  the  arch,  and  that  division  on  the 
nonius  gives  the  odd  minutes  to  be  added  to  that  on  the  arch 
nearest  the  right-hand  of  the  0  on  the  nonius ;  for  example, 
suppose  the  index  division  does  not  coincide  with  25  degrees, 
but  that  the  next  division  to  it  on  the  nonius  is  the  first  coin- 
cident division,  then  is  the  required  angle  25  degrees  1  minute ; 
if  it  had  been  the  second  division  the  angle  would  have  been 
25  degrees  2  minutes,  and  so  on  to  20  minutes,  when  the  0  on 
the  nonius  would  coincide  with  the  first  20  minutes  on  the 
arch  from  25  degrees.  Again,  let  us  suppose  the  0  on  the 
nonius  to  stand  between  50  degrees  and  50  degrees  20  minutes, 
and  that  the  15th  division  on  the  nonius  coincides  with  a 
division  on  the  arch,  then  is  the  angle  50  degrees  15  minutes. 
Further,  let  the  0  on  the  nonius  stand  between  45  degrees 
20  minutes  and  45  degrees  40  minutes,  and  at  the  same  time 
the  14th  division  on  the  nonius  stands  directly  opposite  to  a 


238  THE  QUADRANT. 

division  on  the  arch,  then  will  the  angle  be  45  degrees  34 
minutes. 

The  index  glass  F  is  a  plane  speculum,  or  mirror  of  glass 
quicksilvered,  set  in  a  brass  frame,  and  so  placed  that  the  face 
of  it  is  perpendicular  to  the  plane  of  the  instrument,  and  imme- 
diately over  the  centre  of  motion  of  the  index.  This  mirror 
being  fixed  to  the  index  moves  along  with  it,  and  has  its  direc- 
tion changed  by  the  motion  thereof. 

This  glass  is  designed  to  reflect  the  image  of  the  sun,  or  any 
other  object,  upon  either  of  the  two  horizon  glasses,  from 
whence  it  is  reflected  to  the  eye  of  the  observer.  The  brass 
frame,  with  the  glass,  is  fixed  to  the  index  by  the  screw  'Mj 
the  other  screw  N  serves  to  place  it  in  a  perpendicular  position, 
if  by  any  accident  it  has  been  put  out  of  order. 

The  horizon  glasses  G  and  H  are  two  small  speculums  on 
the  radius  of  the  octant ;  the  surface  of  the  upper  one  is  par- 
allel to  the  index  glass  when  the  0  on  the  nonius  is  at  0  on  the 
arch ;  these  mirrors  receive  the  rays  of  the  object  reflected 
from  the  index  glass,  and  transmit  them  to  the  observer.  The 
fore  horizon  glass  G  is  only  silvered  on  its  lower  half,  the 
upper  half  being  transparent,  in  order  that  the  direct  object 
may  be  seen  through  it.  The  back  horizon  glass  H  is  silvered 
at  both  ends ;  in  the  middle  there  is  a  transparent  slit,  through 
which  the  horizon  may  be  seen.  Each  of  these  glasses  is  set 
in  a  brass  frame,  to  which  there  is  an  axis  ;  this  axis  passes 
through  the  wood-work,  and  is  fitted  to  a  lever  on  the  under  side 
of  the  quadrant,  by  which  the  glass  may  be  turned  a  few  de- 
grees on  its  axis,  in  order  to  set  it  parallel  to  the  index  glass. 

To  set  the  glasses  perpendicular  to  the  plane  of  the  quad- 
rant there  are  two  sunk  screws,  one  before  and  one  behind 
each  glass :  these  screws  pass  through  the  plate  on  which  the 
frame  is  fixed  into  another  plate,  so  that  by  loosening  one  and 
tightening  the  other  of  these  screws,  the  direction  of  the  frame, 
with  its  mirror,  may  be  altered,  and  thus  be  set  perpendicular 
to  the  plane  of  the  instrument. 

The  dark  glasses,  or  shades,  /,  are  used  to  prevent  the  bright 
rays  of  the  sun,  or  the  glare  of  the  moon,  from  hurting  the  eye 
at  the  time  of  observation ;  there  are  generally  three  of  them, 
two  red,  and  one  green.  They  are  each  set  in  a  brass  frame 
which  turns  on  a  centre,  so  that  they  may  be  used  separately 
or  together,  as  the  brightness  of  the  object  may  require.  The 
green  glass  may  be  used  also  alone,  if  the  sun  be  very  faint ; 
it  is  likewise  used  in  taking  observations  of  the  moon ;  when 
these  glasses  are  used  for  the  fore  observation,  they  are  set 


THE  QUADRANT.  230 

immediately  before  the  fore  horizon  glass,  as  in  fig.  1,  but  in 
front  of  the  other  horizon  glass  at  O  when  a  back  observation 
is  made. 

The  sight  vanes  K  and  L  are  pieces  of  brass,  standing  per- 
pendicular to  the  plane  of  the  instrument :  the  vane  K  is  called 
the  fore  sight  vane,  and  L  the  back  sight  vane.  There  are  two 
holes  in  the  fore  sight  vane,  the  lower  of  which  and  the  upper 
edge  of  the  silvered  part  of  the  fore  horizon  glass  are  equi- 
distant from  the  plane  of  the  instrument,  and  the  other  is  oppo- 
site to  the  middle  of  the  transparent  part  of  that  glass ;  the 
back  sight  vane  has  only  one  hole,  which  is  exactly  opposite 
to  the  middle  of  the  transparent  slit  in  the  horizon  glass  to 
which  it  belongs :  but  as  the  back  observations  are  liable  to 
many  inconveniences  and  errors,  we  shall  not  give  any  direc- 
tions for  their  practice. 

The  adjusting  lever  (fig.  3),  which  is  fixed  on  the  back  of 
the  quadrant,  serves  to  adjust  the  horizon  glass,  by  placing  it, 
parallel  to  the  index  glass  ;  when  this  lever  is  to  be  used,  the 
screw  B  must  be  first  loosened,  and  when  by  the  adjuster  <A, 
the  horizon  glass  is  sufficiently  moved,  the  screw  B  must  be 
fastened  again,  by  which  means  the  horizon  glass  will  be  kept 
from  changing  its  position. 

ADJUSTMENTS. 

The  several  parts  of  the  quadrant  being  liable  to  be  out  of 
order  from  a  variety  of  accidental  circumstances,  it  is  neces- 
sary to  examine  and  adjust  them,  so  that  the  instrument  may 
be  pyt  into  a  proper  state  previous  to  taking  observations. 

An  instrument  properly  adjusted  must  have  the  index  glass 
and  horizon  glasses  perpendicular  to  the  plane  of  the  quadrant ; 
the  plane  of  the  fore  horizon  glass  parallel,  and  that  of  the 
back  horizon  glass  perpendicular,  to  the  plane  of  the  index 
glass,  when  the  0  on  the  nonius  is  at  0  on  the  arch  ;  hence, 
the  quadrant  requires  five  adjustments,  the  first  three  of  which, 
being  once  made,  are  not  so  liable  as  the  last  two  to.  be  out  of 
order ;  however,  they  should  all  be  occasionally  examined,  in 
case  of  an  accident 

I.  To  set  the  plane  of  the  index  glass  perpendicular  to  that  of  the  in- 
strument. 

Place  the  index  near  to  the  middle  of  the  arch,  and  holding 
the  quadrant  in  a  horizontal  position,  with  the  index  glass  close 
to  the  eye,  look  obliquely  down  the  glass,  in  such  a  manner 
that  you  may  see  the  arch  of  the  quadrant  by  direct  view  and 
by  reflection  at  the  same  time  ;  if  they  join  in  one  direct  line, 


•*• 


240  THE  QUADRANT. 

and  the  arch  seen  by  reflection  forms  an  exact  plane,  or  straight 
line,  with  the  arch  seen  by  direct  view,  or  if  the  image  of  any 
point  of  the  arch  near  B  appear  of  the  same  height  as  the 
corresponding  part  of  the  arch  near  C,  seen  direct,  the  glass 
is  perpendicular  to  the  plane  of  the  quadrant ;  if  not,  it  must 
be  restored  to  its  right  position  by  loosening  the  screw  M, 
and  tightening  the  screw  JV,  or  vice  versa,  by  a  contrary  ope- 
ration. 

II.  To  set  the  fore  horizon  glass  parallel  to  the  index  glass,  the  index 
leing  at  0. 

Set  the  0  on  the  nonius  exactly  against  0  on  the  arch,  and 
fix  it  there  by  the  screw  at  the  under  side.  Then  holding  the 
quadrant  vertically,  with  the  arch  lowermost,  look  through  the 
sight  vane,  at  the  edge  of  the  sea,  or  any  other  well-defined 
and  distant  object.  Now,  if  the  horizon  in  the  silvered  part 
exactly  meets,  and  forms  one  continued  line  with  that  seen 
through  the  unsilvered  part,  the  horizon  glass  is  parallel  to  the 
index  glass.  But  if  the  horizons  do  not  coincide,  then  loosen 
the  button-screw  in  the  middle  of  the  lever,  on  the  under  side 
of  the  quadrant,  an<1  Jiove  the  horizon  glass  on  its  axis,  by 
turning  the  nut  at  the  end  of  the  adjusting  lever,  till  you  have 
made  them  perfectly  coincide ;  then  fix  the  lever  firmly  in  this 
situation  by  tightening  the  button-screw.  This  adjustment 
ought  to  be  repeated  before  and  after  every  observation.  Some 
observers  adopt  the  following  method,  which  is  called  finding 
the  index  error.  Let  the  horizon  glass  remain  fixed,  and  move 
the  index  till  the  image  and  object  coincide ;  then  observe 
whether  0  on  the  nonius  agrees  with  0  on  the  arch,  if  it  does 
not,  the  number  of  minutes  by  which  they  differ  is  to  be  added 
to  the  observed  altitude  or  angle,  if  the  0  on  the  nonius  be  to 
the  right  of  the  0  on  the  arch,  but  if  to  the  left  of  the  0  on  the 
limb,  it  is  to  be  subtracted. 

It  has  already  been  observed,  that  that  part  of  the  arch  be- 
yond 0  towards  the  right-hand  is  called  the  arch  of  excess  : 
the  nonius,  when  the  0  on  it  is  at  that  part,  must  be  read  the 
contrary  way,  or,  which  is  the  same  thing,  you  may  read  off  the 
minutes  in  the  usual  way,  and  then  their  complement  to  20 
minutes  will  be  the  real  number  to  be  added  to  the  degrees  and 
minutes  pointed  out  by  the  0  on  the  nonius. 

III.  To  set  the  fore  horizon  glass  perpendicular  to  the  plane  of  the 
quadrant. 

Having  previously  made  the  above  adjustment,  incline  the 
quadrant  on  one  side  as  much  as  possible,  provided  the  horizon 
continues  to  be  seen  in  both  parts  of  the  glass ;  if,  when  the 


THE  QUADRANT.  241 

instrument  is  thus  inclined,  the  edge  of  the  sea  seen  through 
the  lower  hole  of  the  sight  vane  continues  to  form  one  unbroken 
line,  the  horizon  glass  is  perfectly  adjusted ;  but  if  the  reflected 
horizon  be  separated  from  that  seen  by  direct  vision,  the  specu- 
lum is  not  perpendicular  to  the  plane  of  the  quadrant :  then  if 
the  limb  of  the  quadrant  is  inclined  towards  the  horizon,  with 
the  face  of  the  instrument  upwards,  and  the  reflected  sea  ap- 
pears higher  than  the  real  sea,  you  must  slacken  the  screw 
before  the  horizon  glass,  and  tighten  that  which  is  behind  it ; 
but  if  the  reflected  sea  appears  lower,  the  contrary  must  be 
performed.  Care  must  be  always  taken  in  this  adjustment  to 
loosen  one  screw  before  the  other  is  screwed  up,  and  to  leave 
the  adjusting  screws  tight,  so  as  to  draw  with  a  moderate  force 
against  each  other. 

This  adjustment  may  be  also  made  by  the  sun,  moon,  or 
a  star :  in  this  case  the  quadrant  is  to  be  held  in  a  vertical  posi- 
tion ;  if  the  image  seen  by  reflection  appears  to  the  right  or 
left  of  the  object  seen  directly,  then  the  glass  must  be  adjusted 
as  before  by  the  two  screws. 

It  will  be  necessary,  after  having  made  this  adjustment,  to 
examine  if  the  horizon  glass  still  continues  to  be  parallel  to 
the  index  glass,  as  sometimes  by  turning  the  sunk  screws  the 
plane  of  the  horizon  glass  will  have  its  position  altered. 

USE  OF  HADLEY'S  QUADRANT. 

The  use  of  the  quadrant  is  to  ascertain  the  angle  subtended 
by  two  distant  objects  at  the  eye  of  the  observer ;  but  princi- 
cipally  to  observe  the  altitude  of  a  celestial  object  above  the 
horizon.  This  is  pointed  out  by  the  index  when  one  of  the 
objects  seen  by  reflection  is  made  to  coincide  with  the  other, 
seen  through  the  transparent  part  of  the  horizon  g'lass. 

To  take  an  altitude  of  the  sun,  moon,  or  a  star,  by  a  fore  observation. 

Having  previously  adjusted  the  instrument,  place  the  0  on 
the  nonius  opposite  to  0  on  the  arch,  and  turn  down  one  or 
more  of  the  screens,  according  to  the  brightness  of  the  sun ; 
then  apply  the  eye  to  the  upper  hole  in  the  fore  sight  vane,  if 
the  sun's  image  be  very  bright,  otherwise  to  the  lower,  and 
holding  the  quadrant  vertically,  look  directly  towards  the  sun. 
so  as  to  let  it  be  behind  the  silvered  part  of  the  horizon  glass, 
then  the  coloured  sun's  image  will  appear  on  the  speculum , 
move  the  index  forward  till  the  sun's  image,  which  will  appear 
to  descend,  just  touches  the  horizon  with  its  lower  or  upper 
limb ;  if  the  upper  hole  be  looked  through,  the  sun's  image 
must  be  made  to  appear  in  the  middle  of  the  trans  arent  part 

L 


242  THE  QUADRANT. 

of  the  horizon,  but  if  it  be  the  lower  hole,  hold  the  quadran 
so  that  the  sun's  image  may  be  bisected  by  the  line  joining  the 
silvered  and  transparent  parts  of  the  horizon  glass.  vj 

The  sun's  limb  ought  to  touch  that  part  of  the  horizon  imme- 
diately under  the  sun,  but  as  this  point  cannot  be  exactly  ascer- 
tained, it  will  be  therefore  necessary  for  the  observer  to  give 
the  quadrant  a  slow  motion  from  side  to  side,  turning  at  the 
same  time  upon  his  heel,  by  which  motion  the  sun  will  appear 
to  sweep  the  horizon,  and  must  be  made  just  to  touch  it  at  the 
lowest  part  of  the  arch ;  the  degrees  and  minutes  then  pointed 
out  by  the  index  on  the  limb  of  the  quadrant  will  be  the  ob- 
served altitude  of  that  limb  which  is  brought  in  contact  with 
the  horizon. 

When  the  meridian  or  greatest  altitude  is  required,  the  ob- 
servation should  be  commenced  a  short  time  before  the  object 
comes  to  the  meridian  ;  being  brought  down  to  the  horizon,  it 
will  appear  for  a  few  minutes  to  rise  slowly ;  when  it  is  again 
to  be  made  to  coincide  with  the  horizon  by  moving  the  index 
forward ;  this  must  be  repeated  until  the  object  begins  to  de- 
scend, when  the  index  is  to  be  secured,  and  the  observation  to 
be  read  off. 

From  this  description  of  the  quadrant  and  its  use,  the  manner  of  adjust- 
ing and  using  the  sextant  will  be  readily  apprehended.  Our  limits  will 
not  allow  a  particular  description  of  this  excellent  instrument. 

The  Artificial  Horizon. 

In  many  cases  it  happens  that  altitudes  are  to  be  taken  on 
land  by  the  quadrant  or  sextant ;  which,  for  want  of  a  natural 
horizon,  can  only  be  obtained  by  an  artificial  one.  There  has 
been  a  variety  of  these  sorts  of  instruments  made,  but  the  kind 
now  described  is  allowed  to  be  the  only  one  that  can  be  de- 
pended upon.  It  consists  of  a  wooden  or  metal-framed  roof, 
containing  two  true  parallel  glasses  of  about  5  by  2|  inches, 
fixed  not  too  tight  in  the  frames  of  the  roof.  This  serves  to 
shelter  from  the  air  a  wooden  trough  filled  with  quicksilver. 
In  making  an  observation  by  it  with  the  quadrant  or  sextant, 
the  reflected  image  of  the  sun,  moon,  or  other  object  is  brought 
to  coincide  with  the  same  object  reflected  from  the  glasses  of 
the  quadrant  or  sextant :  half  the  angle  shown  upon  the  limb 
is  the  altitude  above  the  horizon  or  level  required.  It  is  neces- 
sary in  a  set  of  observations  that  the  roof  be  always  placed 
the  same  way.  When  done  with,  the  roof  folds  up  flatwise, 
and,  with  the  quicksilver  in  a  bottle,  &c.  is  packed  into  a  port- 
able flat  case. 


VARIATION  OF  THE  COMPASS.  243 

SECTION  III. 

VARIATION  OF  THE  COMPASS. 

The  variation  of  the  compass  is  the  deviation  of  the  points 
of  the  mariner's  compass  from  the  corresponding  points  of  the 
horizon,  and  is  termed  east  or  west  variation  according  as  the 
magnetic  needle  or  north  point  of  the  compass  is  inclined  to  the 
eastward  or  westward  of  the  true  north  point  of  the  horizon. 

The  true  amplitude  of  any  celestial  object  is  an  arch  of  the  horizon  con- 
tained between  the  true  east  or  west  points  thereof  and  the  centre  of  the 
object  at  the  time  of  its  rising  or  setting  ;  or  it  is  the  degrees  and  minutes 
the  object  rises  or  sets  to  the  northward  or  southward  of  the  true  east  or 
west  points  of  the  horizon. 

The  magnetic  amplitude  is  an  arch  contained  between  the  east  or  west 
points  of  the  compass  and  the  centre  of  the  object  at  rising  or  setting  ;  or 
it  is  the  bearing  of  the  object  by  compass  when  in  the  horizon. 

The  true  azimuth  of  an  object  is  an  arch  of  the  horizon  contained  be- 
tween the  true  meridian  and  the  azimuth  circle  passing  through  the  cen- 
tre of  the  object. 

The  magnetic  azimuth  is  an  arch  contained  between  the  magnetic  me- 
ridian and  the  azimuth  circle  passing  through  the  centre  of  the  object ;  or 
it  is  the  bearing  of  the  object  by  compass  at  any  time  when  it  is  above  the 
horizon. 

The  true  amplitude  or  azimuth  is  found  by  calculation,  and  the  mag- 
netic amplitude  or  azimuth  by  an  azimuth  compass. 

The  magnetic  amplitude  or  azimuth  of  the  sun,  or  any  celestial  object, 
may  be  accurately  observed  by  Mr.  M'Culloch's  patent  compass,  of  which 
the  following  is  a  description. 

DESCRIPTION   OF   THE   AZIMUTH   COMPASS. 

Frontispiece,  fig.  4,  contains  a  perspective  view  of  the  azi- 
muth compass  ready  for  observation.  The  needle  and  card  of 
this  compass  are  similar  to  those  of  the  steering  compass,  with 
this  difference  only,  that  a  circular  ring  of  silvered  brass,  divided 
into  360°,  or  rather  four  times  90°,  circumscribes  the  card : 
b  represents  the  compass-box,  which  is  of  brass,  and  has  a  hol- 
low conical  bottom  ;  e  is  the  prop  or  support  of  the  compass- 
box,  which  stands  in  a  brass  socket  screwed  to  the  bottom  of 
the  wooden  box,  and  may  be  turned  round  at  pleasure ;  h  is 
one  of  the  guards,  the  other,  being  directly  opposite,  is  hid  by 
the  box,— each  guard  has  a  slit,  in  which  a  pin  projecting  from 
the  side  of  the  box  may  move  freely  in  a  vertical  direction ; 
I  is  a  brass  bar,  upon  which,  at  right  angles,  the  side-vanes  are 
fixed, — a  line  is  drawn  along  the  middle  of  this  bar,  which  line, 
the  lines  in  the  vanes,  and  the  threads  joining  their  tops  are  in 


.  - ^  -*•" 
**'  *' 

244  VARIATION  OF  THE  COMPASS. 

the  same  plane  ;  2  is  a  coloured  glass  moveable  in  the  vane  3 ; 
4  is  a  magnifying  glass  moveable  in  the  other  vane,  whose 
focal  distance  is  nearly  equal  to  the  distance  between  the 
vanes  ;  5  is  the  vernier,  which  contains  six  divisions,  and  as 
the  limb  of  the  card  is  divided  into  half-degrees,  each  division 
of  the  vernier  is  therefore  five  minutes, — the  interior  surface 
of  the  vernier  is  ground  to  a  sphere,  whose  radius  is  equal  to 
that  of  the  card ;  6  is  a  slide  or  stopper  connected  with  the 
vernier,  which  serves  to  push  the  vernier  close  to  the  card, 
and  thereby  prevent  it  from  vibrating  as  soon  as  the  observa- 
tion of  the  amplitude  or  azimuth  is  completed,  and  hence  the 
degrees  and  parts  of  a  degree  may  be  read  off  at  leisure  with 
certainty ;  7  is  a  convex  glass,  to  assist  the  eye  in  reading  off 
the  observed  amplitude  or  azimuth. 

To  observe  the  sun's  amplitude. 

Turn  the  compass-box  until  the  vane  containing  the  magnifying 
glass  is  directed  towards  the  sun ;  and  when  the  bright  speck,  or  rays  of 
the  sun  collected  by  the  magnifying  glass,  falls  upon  the  slit  in  the 
other  vane,  stop  the  card  by  means  of  the  nonius,  and  read  off  the  am- 
plitude. 

Without  using  the  magnifying  glass,  the  sight  may  be  directed  through 
the  dark  glass  towards  the  sun  ;  and  in  this  case  the  card  is  to  be  stopped 
•when  the  sun  is  bisected  by  the  thread  in  the  other  vane. 

The  observation  should  be  made  when  the  sun's  lower  limb  appears 
somewhat  more  than  his  semidiameter  above  the  horizon,  because  his 
centre  is  really  then  in  the  horizon,  although  it  is  apparently  elevated  on 
account  of  the  refraction  of  the  atmosphere  :  this  is  particularly  to  be  no- 
ticed in  high  latitudes. 

To  observe  the  sun's  azimuth. 

Raise  the  magnifying  glass  to  the  upper  part  of  the  vane,  and  move  the 
box,  as  before  directed,  until  the  bright  speck  fall  on  the  other  vane  or  on 
the  line  in  the  horizontal  bar ;  the  card  is  then  to  be  stopped,  and  the 
divisions  being  read  off'  will  be  the  sun's  magnetic  azimuth. 

If  the  card  vibrate  considerably  at  the  time  of  observation,  it  will  be 
better  to  observe  the  extreme  vibrations  and  take  their  mean  as  the  mag- 
netic azimuth.  When  the  magnetic  azimuth  is  observed,  the  altitude  of 
the  object  must  be  taken  in  order  to  obtain  the  true  azimuth. 

It  will  conduce  much  to  accuracy  if  several  azimuths  be  observed,  with 
the  corresponding  altitudes,  and  the  mean  of  the  whole  taken  for  the  ob- 
servation. 

To  find  the  variation  of  the  compass  by  an  amplitude. 

RULE. — 1.  To  the  log.  secant  of  the  latitude,  rejecting  the 
index,  add  the^log.  sine  of  the  sun's  declination,  corrected  for 
the  time  and  place  of  observation;  their  sum  will  be  the  log.! 
sine  of  the  true  amplitude,  to  be  reckoned  from  the  east  in  the  4 


VARIATION  OF  THE  COMPASS.  245 

morning  or  the  west  in  the  afternoon,  towards  the  north 'or  south, 
according  to  the  declination. 

2.  Then  if  the  true  and  magnetic  amplitudes  be  both  north 
or  both  south  their  difference  is  the  variation,  but  if  one  be 
north  and  the  other  south  their  sum  is  the  variation ;  and  to 
know  whether  it  be  easterly  or  westerly,  suppose  the  observer 
looking  towards  that  point  of  the  compass  representing  the 
magnetic  amplitude ;  then  if  the  true  amplitude  be  to  the  right- 
hand  of  the  magnetic  amplitude  the  variation  is  east,  but  if  to 
the  left-hand  it  is  west. 

EXAMPLE    I. 

July  3,  1812,  in  latitude  9°  36'  S.  the  sun  was  observed  to  rise  E.  12° 
42'  Nt ;  required  the  variation  of  the  compass. 

Latitude  9°36'S.    -    -     Secant  0.00613 

Declination  22   59  N.    -    -     Sine     9.50158 


True  amplitude  E.  23    20  N.    -    -     Sine      9.59771 
Mag.  amplitude  E.  12   42  N. 

Variation  10   38  W.,  because  the  true  amplitude  is  to 

the. left  of  the  magnetic. 

EXAMPLE    II. 

September  24,  1812,  in  latitude  26°  32'  N.  and  longitude  78°  W.  the 
sun's  centre  was  observed  to  set  W.  60  15'  S.  about  6h.  P.  M. ;  required 
the  variation  of  the  compass. 

Sun's  declination  0°  30*  S. 

Corr.  for  long.  78°  W.  4-   5 
Corr.  for  time  6h.  P.  M.  +   6 

Reduced  declination       0   41     -    -    -     Sine        8.07650 
Latitude  26   32    -    -    -     Secant    0.04834 

True  amplitude      W.    0   46  S.    -    -      Sine       8.12484 
Mag.  amplitude      W.    6    15  S. 

Variation  5    29  E.,  because  the  true  amplitude  is  to 

the  right-hand  of  the  magnetic. 

To  find  the  variation  of  the  compass  by  an  azimuth. 

RULE. — 1.  Reduce  the  sun's  declination  to  the  time  and  place 
of  observation,  and  compute  the  true  altitude  of  the  sun's  centre. 

2.  Subtract  the  sun's  declination  from  90°  when  the  latitude 
and  decimation  are  of  the  same  name,  or  add  it  to  90°  when 
they  are  of  contrary  names,  and  the  sum  or  remainder  will 
be  the  sun's  polar  distance. 

3.  Add  together  the  sun's  polar  distance,  the*latitude  of  the 
place,  and  the  altitude  of  the  sun ;  take  the  difference  between 
half  their  sum  and  the  polar  distance,  and  note  the  remainder. 


246  VARIATION  OF  THE  COMPASS> 

4.  Then  add  together 

the  log.  secant  of  the  altitude  )     .     ..      .,    -    -   ••  ^ 

r  i     T    •    j    ?  rejecting  their  indices, 
the  log.  secant  of  the  latitude  )     J 

the  log.  co-sine  of  the  half-sum, 
and  the  log.  co-sine  of  the  remainder. 

5.  Half  the  sum  of  these  four  logarithms  will  be  the  sine 
of  an  arch,  which  doubled  will  be  the  sun's  true  azimuth ;  to 
be  reckoned  from  the  south  in  north  latitude,  and  from  the  north 
in  south  latitude ;  towards  the  east  in  the  morning,  and  towards 
the  west  in  the  afternoon. 

6.  Then  if  the  true  and  observed  azimuths  be  both  on  the 
east  or  both  on  the  west  side  of  the  meridian,  their  difference 
is  the  variation ;  but  if  one  be  on  the  east  and  the  other  on  the 
west  side  of  the  meridian,  their  sum  is  the  variation :  and  to 
know  if  it  be  east  or  west,  suppose  the  observer  looking  to- 
wards that  point  of  the  compass  representing  the  magnetic 
azimuth ;  then  if  the  true  azimuth  be  to  the  right  of  the  mag- 
netic, the  variation  is  east,  but  if  the  true  be  to  the  left  of  the 
magnetic  the  variation  is  west. 

EXAMPLE. 

November  2,  1812,  in  latitude  25°  32'  N.  and  longitude  75° 
W.  the  altitude  of  the  sun's  lower  limb  was  observed  to  be  15° 
36',  about  4h.  10m.  P.  M.,  his  magnetic  azimuth  at  that  time 
being  S.  58°  32'  W.,  and  the  height  of  the  eye  18  feet ;  re- 
quired the  variation  of  the  compass. 

Sun's  dec.  Nov.  2,  at  n.  14°  48'  S.   Obs.  alt.  sun's  lower  limb  15°  36* 
Com  for  long.  75°  W.      +      4        Semidiameter  16' >  ,    .„ 

Co.  for  ti.  4h.  10m.  af.  n.  -(-      3        Dip  4  J 

Reduced  declination          14    55 

90    00        Refraction 

Polar  distance          104    55        True  altitude  15   45 

Altitude  15    45        -    -    Secant  0.01662 

Latitude  25    32       -    -    Secant  0.04463 

Sum 

Half  73      6       -    -    Co-sine  9.46345 

Remainder  31    49       -    -    Co-sine  9.92929 


32    14       -    -    Sine     9.72699 


True  azimuth  S.  64  28  W. 

Mag.  azimuth  S,  68  32  W. 

Variation  5  56  east,  because  the  true  azimuth  is  to  the 
light  of  the  magnetic. 


VARIATION  OF  THE  COMPASS.  247j 

To  draw  a  true  meridian  line  to  a  map,  having  the  variation  and  mag- 
netical  meridian  given. 

On  any  magnetical  meridian  or  parallel,  upon  which  the  map  is  pro- 
tracted, set  off  an  angle  from  the  north  towards  the  east,  equal  to  the  de- 
grees or  quantity  of  variation  if  it  be  westerly,  or  from  the  north  towards 
the  west  if  it  be  easterly,  and  the  line  which  constitutes  such  an  angle 
with  the  magnetical  meridian  will  be  a  true  meridian  line. 

For  if  the  variation  be  westerly,  the  magnetical  meridian  will  be  the 
quantity  of  vanation  of  the  west  side  of  the  true  meridian,  but  if  easterly, 
on  the  east  side  ;  therefore  the  true  meridian  must  be  a  like  quantity  on 
the  east  side  of  the  magnetical  one  when  the  variation  is  westerly,  and 
on  the  west  side  when  it  is  easterly. 

To  lay  out  a  true  meridian  line  by  the  circumferentor. 

If  the  variation  be  westerly,  turn  the  box  about  till  the  north  of  the  needle 
points  as  many  degrees  from  the  flower-de-luce  towards  the  east  of  the 
box,  or  till  the  south  of  the  needle  points  the  like  number  of  degrees  from 
the  south  towards  the  west,  as  are  the  number  of  degrees  contained  in  the 
variation,  and  the  index  will  be  then  due  north  and  south  ;  therefore,  if  a 
line  be  struck  out  in  the  direction  thereof,  it  will  be  a  true  meridian  line. 

If  the  variation  was  easterly,  let  the  north  of  the  needle  point  as  many 
degrees  from  the  flower-de-luce  towards  the  west  of  the  box,  or  let  the 
south  of  the  needle  point  as  many  degrees  towards  the  east,  as  are  the 
number  of  degrees  contained  in  the  variation,  and  then  the  north  and  south 
of  the  box  will  coincide  with  the  north  and  south  points  of  the  horizon, 
and  consequently  a  line  being  laid  out  by  the  direction  of  the  index  will 
be  a  true  meridian  line. 

This  will  be  found  to  be  very  useful  in  setting  a  horizontal  dial,  for  if 
you  lay  the  edge  of  the  index  by  the  base  of  the  stile  of  the  dial,  and  keep 
the  angular  point  of  the  stile  towards  the  south  of  the  box,  and  allow  the 
variation  as  before,  the  dial  will  then  be  due  north  and  soufh,  and  in  its 
proper  situation,  provided  the  plane  upon  which  it  is  fixed  be  duly  hori- 
zontal, and  the  sun  be  south  at  noon  ;  but  in  places  where  it  is  north  at 
noon  the  angular  point  of  the  index  must  be  turned  to  the  north. 

How  maps  may  be  traced  by  the  help  of  a  true  meridian  line. 

If  all  maps  had  a  true  meridian  line  laid  out  upon  them,  it  would  be 
easy,  by  producing  it,  and  drawing  parallels,  to  make  out  field-notes ; 
and  by  knowing  the  variation,  and  allowing  it  upon  every  bearing,  and 
having  the  distances,  you  would  have  notes  sufficient  for  a  trace.  But 
a  true  meridian  line  is  seldom  to  be  met  with ;  therefore  we  are  obliged  to 
have  recourse  to  the  foregoing  method.  It  is  therefore  advised  to  lay  out 
a  true  meridian  line  upon  every  map. 

To  find  the  difference  between  the  present  variation,  and  that  at  a  time 
when  a  tract  was  formerly  surveyed,  in  order  to  trace  or  run  out  the  original 
lines. 

If  the  old  variation  be  specified  in  the  map  or  writings,  and  the  present 
be  known,  by  calculation  or  otherwise,  then  the  difference  is  immediately 
seen  by  inspection  ;  but  as  it  more  frequently  happens  that  neither 
is  certainly  known,  and  as  the  variation  of  different  instruments  is  not 
always  alike  at  the  same  time,  the  following  practical  method  will  be  found 
to  answer  every  purpose. 


248  VARIATION  OF  THE  COMPASS. 

Go  to  any  part  of  the  premises  where  any  two  adjacent  corners  are 
known ;  and  if  one  can  be  seen  from  the  other,  take  their  bearing ;  which, 
compared  with  that  of  the  same  line  in  the  former  survey,  shows  the  dif- 
ference. But  if  trees,  hills,  &c.  obstruct  the  view  of  the  object,  run  the 
I  line  according  to  the  given  bearing,  and  observe  the  nearest  distance  be- 
,'tween  the  line  so  run  and  the  corner,  then, 

As  the  length  of  the  whole  line 

Is  to  57.3  degrees,* 

So  is  the  said  distance 

To  the  difference  of  variation  required. 

EXAMPLE. 

Suppose  it  be  required  to  run  a  line  which  some  years  ago  bore  NE.  45°, 
^distance  80  perches,  and  in  running  this  line  by  the  given  bearing,  the 
corner  is  found  20  links  to  the  left-hand  ;  what  allowance  must  be  made  on 
each  bearing  to  trace  the  old  lines,  and  what  is  the  present  bearing  of  this 
particular  line  by  the  compass  1 

P.  Deg.  L. 

As  80        :         57.3     :     :       20 
25  20 


21000  1 146.0(0  °34f 

60 


2)68  [760.0 

Answer,  34  minutes,  or  a  little  better  than  half  a  degree  to  the  left- 
hand,  is  the  allowance  required,  and  the  line  in  question  bears  N.  44°  26'E. 

Note. — The  different  variations  do  not  affect  the  area  in  the  calculation, 
as  they  are  similar  in  every  part  of  the  survey. 

*  57.3  is  the  radius  of  a  circle  (nearly)  in  such  parts  as  the  circumference  contains  360. 


THE   END 


Harper's  Stereotype  Edition., 


TABLE    OF   LOGARITHMS, 


OF 


LOGARITHMIC    SIXES, 


AND   A 


TRAVERSE    TABLE. 


DESCRIPTION 


OF 


THE    TABLES. 


1.  LOGARITHMS  of  numbers  are  the  indices  that  denote  the 
different  powers  to  which  a  given  number  must  be  raised  to 
produce  those  numbers. 

2.  If  a  be  the  given  number,  whose  indices  and  powers  are 
to  be  considered,  then  a±*  being  put  equal  to  n,  a,  the  given 
number,  or  root,  is  called  the  base  of  the  system  of  logarithms, 
n  the  number  whose  logarithm  is  considered,  and  ±a?,  the  loga- 
rithm of  that  number. 

3.  Any  number,  except  1,  may  be  taken  for  the  base  of  a 
system  of  logarithms.     In  the  system  in  general  use,  the  base 
is  10 ;  and  this  system  affords  the  greatest  facilities  in  calcula- 
tions, because  10  is  the  base  of  the  common  numeration,  both  in 
whole  numbers  and  decimal  fractions.  ,    > 

4.  Taking  a±z=n,  we  have,  ±a:=log.  n ;  and  putting  a±^= 
OT,  gives,  ±y=log.  m.     If  the  equations,  az=7»,  and  ay=-m,  be 
multiplied  together,  member  by  member,  we  have,  a*Xay= 
nXwi?  or  az+y=7iXwi.     In  this  expression,  x-{-y  is  the  loga- 
rithm of  nXm  (2) ;  from  which  we  conclude,  that  the  sum  of 
the  logarithms  of  any  two  numbers,  is  equal  to  the  logarithm  of 
their  product. 

5.  If  the  equations  oz=n,  c^=wi,  be  divided,  member  by 

member,  — =-;  or  az—y=-.    In  this  expression,  a? — y  is  the 
cP     m  m 

logarithm  of  -  (2) ;  from  which  we  conclude,  that  the  dijfer- 
m 

ence  of  the  logarithms  of  any  two  numbers^  is  equal  to  t\e  logo 
rithm  of  their  quotient. 


4  DESCRIPTION    OF 

6.  If  in  the  equation  a*=ra,  both  members  be  raised  to  the 
m\h  power,  amx  =  nm.     Here,  mx  is  the  logarithm  of  nm  ;  from 
which  it  appears,  that  the  logarithm  of  the  power  of  any  number, 
is  equal  to  the  logarithm  of  that  number,  multiplied  by  the  index 
of  that  power. 

7.  If  the  mth  root  of  both  members  of  the  equation  ax—n, 

•"!.«.  I 

be  taken,  then,  am=nm;  but-  is  the  logarithm  of  nm;  from 

m 

which  it  appears,  that  the  logarithm  of  the.  root  of  any  number, 
is  equal  to  the  logarithm  of  that  number  divided  by  the  index  of 
the  root. 

8..  It  is  evident,  that  the  results  obtained  in  the  last  four  arti- 
cles are  equally  true,  whether  the  logarithms  be  positive  or  nega- 
tive. These  results  show,~that  the  addition  of  logarithms  cor- 
responds to  the  multiplication  of  their  numbers  ;  the  subtraction 
of  logarithms,  to  the  division  of  numbers  ;  their  multiplication, 
to  the  raising  of  powers  ;  and  their  division,  to  the  extraction  of 
roots. 

9.  Returning  to  the  equation  a±x—n,  in  which  ±a?=log.  n, 
and  applying  it  to  the  common^  system,  in  which  the  base  is 
10,  we  have, 

(10)4  :  (10)3  :  (10)2  .  (10)i  :  (10)°  :  (10)-1    (10)~2  :  (10)-3  :  (10) 
10000:1000:100:    10    :    1      :  0.1 


4:3      :    2    :     1     :    0     :  —  1 


0.01     :  0.001    :  0.0001  num 
—  2      :  —  3      :  —  4     log. 


Unity  being  the  number  which  divides  the  whole  numbers 
from  the  decimal  fractions,  we  shall  begin  with  it,  and  explain 
some  properties  of  the  logarithms  of  whole  numbers.  The 
logarithm  of  1  is  0  ^  and  this  is  the  case  in  all  systems,  for 
whatever  be  the  base,  its  0  power  is  1  :  but  the  index  of  the 
base  is  the  logarithm  of  the  power  ;  therefore,  0  is  the  loga- 
rithm of  1.  As  the  logarithms  increase  with  the  numbers  from 
unity  upwards,  tne  logarithms  of  all  numbers,  which  are  greater 
than  1,  and  less  than  10,  are  greater  than  0,  and  less  than  1  : 
their  values  are  generally  expressed  by  decimal  fractions  ;  thus, 
the  log.  2=0.301030.  The  logarithms  of  numbers  greater 
than  10,  and  less  than  100,  lie  between  1  and  2,  and  are  gene- 
rally expressed  by  unity  and  a  decimal  fraction  :  thus,  the  log. 
50  =  1.698970. 

The  logarithms  of  numbers  greater  than  100,  but  less  than 
1000,  are  greater  than  2,  and  less  than  3,  and  are  expressed 


•»-  .  •  • 

THE    TABLES.  0 

by  uniting  2  with  a  decimal  fraction:  thus,  the  log.  126  = 
2.100371.  The  whole  number  on  the  left  of  the  decimal 
point  is  called  the  characteristic,  or  index  of  the  logarithm. 
The  number  of  units  which  it  contains,  is  always  one  less  than 
the  number  of  places  of  figures  in  the  number  whose  logarithm  is 
taken.  Thus,  in  the  first  case,  for  numbers  between  1  and  10, 
there  is  but  one  place  of  figures,  and  the  characteristic  is  0. 
In  the  second  case,  for  numbers  between  10  and  100,  there  are 
two  places,  and  the  characteristic  is  1.  In  the  third  case,  for 
numbers  between  100  and  1000,  there  are  three  places,  and 
the  characteristic  is  2 ;  and  in  like  manner  for  any  number  of 
places  whatsoever. 

TABLE  OF  LOGARITHMS. 

10.  If  the  logarithms  of  all  the  numbers  between  1  and  any 
given  number,  be  calculated  and  arranged  in  a  tabular  form, 
such  table  is  called  a  table  of  logarithms.     The  table  annexed 
shows  the  logarithms  of  all  numbers  between  1  and  10,000. 

11.  The  first  column,  on  the  left  of  each  page  of  the  table 
of  logarithms,  is  the  column  of  numbers,  and  is  designated  by 
the  letter  N;   the  logarithms  of  these  numbers   are  placed 
directly  opposite  them,  and  on  the  same  horizontal  line. 

12.  To  find,  from   the  table,  the   logarithm  of  any   whole 
number. 

If  the  number  be  less  than  100,  look  on  the  first  page  of  the 
table  of  logarithms,  along  the  columns  of  numbers  under  N, 
until  the  number  is  found ;  the  number  directly  opposite  it,  in 
the  column  designated  Log.,  is  the  logarithm  sought. 

13.  When  the  number  is  greater  than  100,  and  less  than  10,000. 
Find,  in  the  column  of  numbers,  the  first  three  figures  of  the 

given  number.  Then,  pass  across  the  page,  in  a  horizontal 
line,  into  the  columns  marked  0,  1,  2,  3,  4,  &c.,  until  you  come 
to  the  column  which  is  designated  by  the  fourth  figure  of  the 
given  number:  to  the  four  figures  so  found,  two  figures  taken 
from  the  column  marked  0,  are  to  be  prefixed.  If  the  first 
four  figures  found  stand  opposite  to  a  row  of  six  figures  in  the 
column  marked  0,  the  two  figures  from  this  column,  which  are 
to  be  prefixed  to  the  four  before  found,  are  the  first  two  on  the 


6  x         DESCRIPTION   OF 

left  hand ;  but,  if  the  first  four  figures  are  opposite  a  line  of 
only  four  figures,  you  are  then 'to  ascend  the  column,  till  you 
come  to  the  line  of  six  figures :  the  two  figures  at  the  left  hand 
are  to  be  prefixed,  and  then  the  decimal  part  of  the  logarithm 
is  obtained ;  to  which  prefix  the  characteristic  (9),  and  you  have 
the  logarithm  sought.  In  several  of  the  column?  designated 
0,  1,  2,  3,  4,  5,  &c.,  small  dots  are  found.  In  such  cases,  a 
cipher  must  be  written  for  each  of  those  dots ;  and  the  two 
figures,  from  the  first  column,  which  are  to  be  prefixed,  are 
found  in  the  horizontal  line  directly  below.  Thus,  the  log.  2188 
is  3.340047,  the  two  dots  being  changed  into  two  ciphers,  and 
the  34  from  the  column  0,  prefixed.  The  two  figures  from  the 
column  0,  must  also  be  taken  from  the  line  below,  if  any  dots 
shall  have  been  passed  over,  in  passing  along  the  horizontal 
line :  thus,  the  logarithm  of  3098  is  3.491081,  the  49  from  the 
column  0  being  taken  from  the  line  310. 

14.  If  the  number  exceeds  10,000,  or  consists  of  fve  or  more 
places  of  figures,  consider  all  the  figures  after  the  fourth  from 
the  left  hand,  as  ciphers.  Find,  from  the  table,  the  logarithm 
of  this  number,  which  will  be  the  same  as  the  logarithm  of  the 
first  four  places,  excepting  the  characteristic.  Take  from  the 
last  column  on  the  right  of  the  page,  marked  D,  the  number  on 
the  same  horizontal  line  with  the  logarithm,  and  multiply  this 
number  by  the  numbers  that  have  been  considered  as  ciphers: 
then,  cut  off  from  the  right-hand  as  many  places  for  decimals 
as  there  are  figures  in  the  multiplier,  and  add  the  product,  so 
obtained,  to  the  first  logarithm,  for  the  logarithm  sought. 

Let  it  be  required  to  find  the  logarithm  of  672887.  The 
log.  of  672800  is  found,  on  the  llth  page  of  the  table,  to  be 
5.827886,  by  prefixing  the  characteristic  5.  The  number  cor- 
responding in  the  column  D  is  65,  which  being  multiplied  by 
87,  the  figures  regarded  as  ciphers,  gives  5655 ;  then,  pointing 
off  two  plaees  for  decimals,  the  number  to  be  added  is  56.55. 
This  number  being  added  to  5.827886,  gives  5.827942  for  the 
logarithm  of  672887 ;  the  decimal  part,  .55,  being  omitted. 

This  method  of  finding  the  logarithms  of  numbers  from  the 
table,  supposes  that  the  logarithms  are  proportional  to  their 
respective  numbers,  which  is  not  rigorously  true.  In  the 
example,  the  logarithm  of  672800  is  5.827886 ;  of  672900, 
a  number  greater  by  100,  5.827951 :  the  difference  of  the 


THE    TABLES.  7 

logarithms  is  65.  Now,  as  100,  the  difference  of  the  numbers^ 
is  to  65,  the  difference  of  their  logarithms,  so  is  87,  the  differ- 
ence between  the  given  number  and  the  least  of  the  numbers 
used,  to  the  difference  of  their  logarithms,  which  is  56.55  : 
this  difference  bfeing  added  to  5.827886,  the  logarithm  of  the 
less  number,  gives  5.827942  for  the  logarithm  of  672887. 
The  use  of  the  column  of  differences  is  therefore  manifest. 

15.  The  logarithm  of  a  fractional  number  is  easily  found, 
from  what  has   already  been  said.     If  the  fractional  number 
exceeds  unity,  as  y/,  its  logarithm  is  equal  to  the  log.  136  — 
log.  25  (5).     If  it  be  less  than  unity,  as  Ty^,  its  logarithm  may 
be  written  under  two  different  forms.     First,  the  log.  j-/j  = 
log.   15—  log.  125  —  —  (log.  125—  log,   15)=  —(2.096910  — 
1.176091)  =  —  0.920819;  the  number  0.920819  being  entirely 
negative.     In  the  equation  log.  15—  log.  125  =  —  0.920819,  if 
the  log.  125  be  transposed  to  the  second  member,  the  log.  15 
=log.  125—0.920819.      Let  N'  be  the  number  whose  loga- 
rithm is   —  0.920819,  and  N  the  number  whose  logarithm  is 
-f  0.920819  ;  then,  the  log.  15—  log.  125=log.  N'.     Since  the 
difference  of  logarithms  of  the  two  numbers  is  equal  to  the 

125 

logarithm  of  their  quotient  (5),  the  log.  15=log.  —  -.     But  if 

the  logarithms  are  equal,  the  numbers  themselves  are  equal  ; 

therefore,  15=——,  or  —  =—  =N',  since  —  is  the  number 
N         125     N  125 

whose  logarithm  is  —  0.920819.  As  the  same  reasoning 
holds  true  for  any  numbers  whatever,  we  conclude,  that 
the  number  answering  to  a  negative  logarithm,  is  the  reciprocal 
of  the  number  answering  to  this  same  logarithm  regarded  as 
positive. 

16.  To  find  the  logarithm  of  a  proper  fraction  under  another 
form.     Let  the  fraction  be  N  =  T1F225T.     Let  this  fraction  be 
multiplied  by  10,  100,  1000,  10,000,  or  such  higher  power  of 
10,  as  to  make  it  greater  than  unity.     If  it  be  multiplied  by 


10,000,  we  shall  have,  10,OOON=  —!9  and  taking  the 

oo27 

logarithms,  4+  log.  N=  4  -flog.  125  -log.  5627  =4  -f-  2.096910 
—3.750277  =  6.096910  —  3.750277=2.346633:  hence  the 
log.  N  =2.346633—  4=2".346633,  the  minus  sign  belonging  to 


8  DESCRIPTION   OF 

*v  X 

the  characteristic  only,  and  not  to  the  decimal  part  of  the  loga- 
rithm. In  such  case,  the  minus  sign  is  written  above  the 
number ;  thus,  2.  If,  then,  it  be  required  to  express  the  loga- 
rithm of  a  fractional  number,  under  such  ^  form  that  the 
characteristic  only  shall  be  negative,  add  such  a  whole  number 
to  the  logarithm  of  the  numerator,  as  will  make  it  greater  than 
the  logarithm,  of  the  denominator ;  from  this  sum,  subtract  the 
logarithm  of  the  denominator,  and  from  the  remainder,  the  whole 
number  which  was  added  to  the  logarithm  of  the  numerator :  the 
remainder  is  the  logarithm  sought. 

17.  To  find  the  logarithm  of  a  decimal  number.     If  the  num- 
ber be  composed  of  a  whole  number  and  a  decimal,  such  as, 
36.78,  it  may  be  put  under  the  form  \<Ly  :  the  log.  WJ5  = 
log.  3678—2  =  3.565612—2  =  1.565612;  from  which  we  see, 
that  the  mixed  number  may  be  treated  as  a  whole  number,  except 
in  fixing  the  value  of  the  characteristic,  which  is  one  less  than 
the  number  of  places  on  the  left  of  the  decimal  point. 

18.  The  logarithm  of  a  decimal  fraction  is  also  readily  found. 
The  log.  0.8=log.  T\=log.  8— 1  =  — 1+log.  8.     Now  the 
log.  8  is  0.903090,  which  is  positive,  and  less  than  1 ;  hence 
log.  0.8=1.903090,  where  the  minus  sign  belongs  to  the  cha- 
racteristic only  (16) ;  hence,  it  appears,  that  the  logarithms  of 
tenths,  are  the  same  as  the  logarithms  of  the  corresponding  whole 
numbers,  excepting,  that  the  characteristic  instead  of  being  0,  is 
—  1.     If  the  fraction  were  of  the  form  .06,  it  might  be  written 
T°/o  ;   taking  the  logarithms,  log.  TYtr=log-  06— 2  =  — 2-f- 
log.  06.     Now,  the  log.  06  is  but  the  log.  6  ;  therefore,  the  log. 
06=2.778151,  the  minus  sign  belonging  only  to  the  character- 
istic, the  decimal  part  being  positive  (16).     If  the  decimal  were 
.006,  its  logarithm  would  be  the  same,  excepting  the  charac- 
teristic, which  would  be  — 3.     It  is,  indeed,  evident,  that  the 
negative  characteristic  will  always  be  one  greater  than  the 
number  of  ciphers  between  the  decimal  point  and  the  first  sig- 
nificant place  of  figures ;  therefore,  the  logarithm  of  a  decimal 
fraction  is  found,  by  considering  it  as  a  whole  number,  and  then 
prefixing  to  its  logarithm  a  negative  characteristic,  greater  by 
unity  than  the  number  of  ciphers  between  the  decimal  point  and 
the  first  significant  place  of  figures. 


THE    TABLES.  9 

19.  To  find,  in  the  table,  a  number  answering  to  a  given 
logarithm. 

Search,  in  the  column  of  logarithms,  for  the  decimal  part  of 
the  given  logarithm,  and  if  it  be  exactly  found,  set  down  the 
corresponding  number.  Then,  if  the  characteristic  of  the 
given  logarithm  be  positive,  point  off,  from  the  left  of  the  num- 
ber found,  one  place  more  for  whole  numbers  than  there  are 
units  in  the  characteristic  of  the  given  logarithm,  and  treat  the 
other  places  as  decimals :  tins  will  be  the  logarithm  sought  (9). 
If  the  characteristic  of  the  given  logarithm  be  0,  there  will  be 
one  place  of  whole  numbers ;  if  it  be  — 1,  the  number  will  be 
entirely  decimal ;  if  it  be  — 2,  there  will  be  one  cipher  between 
the  decimal  point  and  the  first  significant  figure ;  if  it  be  — 3, 
there  will  be  two,  &e.  The  number  whose,  logarithm  is 
1.492481  is  found  in  page  5,  and  is  .31.08. 

But  if  the  decimal  part  of  the  logarithm  cannot  be  exactly 
found  in  the  table,  take  the  number  answering  to  the  next  less 
logarithm ;  take  also  from  the  table  the  corresponding  differ- 
ence in  the  column  D :  then,  subtract  this  less  logarithm  from 
the  given  logarithm;  divide  the  remainder  by  the  difference 
taken  from  'the  column  D,  and  annex  the  quotient  to  the  number 
answering  to  the  less  logarithm :  this  gives  the  required  number, 
nearly.  This  rule,  like  the  one  for  finding  the  logarithm  of  a 
number  when  the  places  exceed  four,  supposes  the  numbers  to 
be  proportional  to  their  corresponding  logarithms.  *v 

Ex.  1.  To  find  the  number  answering  to  the  logarithm 
1.532708.  Here, 

Next  less  log.  is  1.532627,  its  number  34.09,  the  tab.  diff.  128. 
The  difference  between  the  given  log.  1.532708  and  1.532627 
is  81 ;  therefore,  128)  8100  (63 

which,  being  decimals  of  a  unit,  in  respect  of  the  9  in  the  num- 
ber 34.09,  must  be  annexed,  and  being  so  annexed,  gives 
34.0963  for  the  number  answering  to  the  log.  1.532708. 

Ex.  2.  Required  the  number  answering  to  the  logarithm 
3.233568. 

The  given  logarithm  =  3.233568 
The  next  less  tabular  logarithm  of  1712  =  3.233504 


Diff.  =  64 

Tab.  Diff.  =  253)  64.00  (25 
2 


10  DESCRIPTION   OF 

Hence  the  number  sought  is  1712.25,  marking  four  places 
of  integers  for  the  characteristic  3. 


TABLE  OF  LOGARITHMIC  SINES. 

20.  In  this  table  are  arranged  the  logarithms  of  the  numeri- 
cal values  of  the  sines,  cosines,  tangents,  and  cotangents,  of  all 
the  arcs  or  angles  of  the  quadrant,  divided  to  minutes,  and  cal- 
culated for  a  radius  of  10,000,000,000.     The  logarithm  of  this 
radius  is  10  (9).     In  the  first  and  last  horizontal  line  of  each 
page,  are  Written  the  degrees  whose  logarithmic  sines,  &c.  are 
expressed  on  the  page.     The  vertical  columns  on  the  left  and 
right,  are  columns  of  minutes. 

21.  To  find,  in  the  table,  the  logarithmic  sine,  cosine,  tangent, 
or  cotangent  of  any  given  arc  or  angle. 

1.  If  the  angle  be  less  than  45°,  look  in  the  first  horizontal 
line  of  the  different  pages,  until  the  number  of  degrees  be  found ; 
then  descend  along  the  column  of  minutes,  on  the  left  of  the 
page,  till  you  reach  the  number  showing  the  minutes ;  then 
pass  along  the  horizontal  line  till  you  come  into  the  column 
designated,  sine,  cosine,  tangent,  or  cotangent,  as  the  case  may 
be :  the  number  so  indicated,  is  the  logarithm  sought.     Thus, 
the  sine,  cosine,  tangent,  and  cotangent  of  19°  55',  are  found  on 
page  37,  opposite  55,  and  are,  respectively,  9.532312,  9.973215, 
9.559097,  10.440903. 

2.  If  the  angle  be  greater  than  45°,  search  along  the  bottom 
line  of  the  different  pages,  till  the  number  of  degrees  are  found ; 
then  ascend  along  the  column  of  minutes,  on  the  right-hand  side 
of  the  page,  till  you  reach  the  number  expressing  the  minutes ; 
then  pass  along  the  horizontal  line  into  the  columns  designated 
tang.,  cotang.,  sine,  cosine,  which  correspond  to  the  degrees 
indicated  at  the  bottom  of  the  page ;  the  number  so  pointed 
out  is  the  logarithm  required. 

22.  It  will  be  seen,  that  the  column  designated  sine  at  the 
top  of  the  page,  is  designated  cosine  at  the  bottom ;  the  one 
designated  tang.,  by  cotang. ;  and  the  one  designated  cotang., 
by  tang. 

The  angle  found  by  taking  the  degrees  at  the  top  of  the  page, 


THE   TABLES.  11 

and  the  minutes  from  the  first  vertical  column  on  the  left,  is  the 
complement  of  the  angle,  found  by  taking  the  corresponding 
degrees  at  the  bottom  of  the  page,  and  the  minutes  traced  up 
in  the  right-hand  column  to  the  same  horizontal  line.  This 
being  apparent,  the  reason  is  manifest,  why  the  columns  desig- 
nated sine,  cosine,  tang.,  and  cotang.,  when  the  degrees  are 
pointed  out  at  the  top  of  the  page,  and  the  minutes  counted 
downwards,  ought  to  be  changed,  respectively,  into  cosine, 
sine,  cotang.,  and  tang.,  when  the  degrees  are  shown  at  the 
bottom  of  the  page,  and  the  minutes  counted  upward. 

23.  If  an  angle  be  greater  than  90°,  we  have  only  to  sub- 
tract it  from  180°,  and  take  the  sine,  cosine,  tangent,  or  cotan- 
gent of  the  remainder. 

24.  The  secants  and  cosecants  are  omitted  in  the  table,  being 
easily  found  from  the  cosines  and  sines. 

R2 

For,  sec.= ;    or,  taking  ihe  logarithms,  log.  sec.  =  2 

log.  R — log.  cos.  =20  — log.  cos.;  that  is,  the  logarithmic 
secant  is  found  by  subtracting  the  logarithmic  cosine  from  20. 

R2 

Andcosec.  =- — ,  or  log.  cosec.  =2  log.  R— log.  sine  =20 
sine 

— log.  sine ;  that  is,  the  logarithmic  cosecant  is  found  by  sub- 
tracting the  logarithmic  sine  from  20. 

It  has  been  shown  that  R2=tang.  X  cotang;  therefore,  2 
log.  R.=log.  tang.  +  log.  cotang;  or,  20=log.  tang. -flog, 
cotang. 

25.  The  column  of  the  table,  next  to  the  column  of  sines, 
and  on  the  right  of  it,  is  designated  by  the  letter  D.     This 
column  is  calculated  in  the  following  manner.     Opening  the 
table   at  any  page,   as  42,  the  sine  of   24°  is  found  to  be 
9.609313;  of  24°  1',  9.609597:  their  difference  is  284;  this 
being  divided  by  60,  the  number  of  seconds  in  a  minute,  gives 
4.73,  which  is  entered  in  the  column  D,  omitting  the  decimal 
point.     Now,  supposing  the  increase  of  the  logarithmic  sine  to 
be  proportional  to  the  increase  of  the  arc,  and  it  is  nearly  so 
for  60",  it  follows,  that  473  (the  last  two  places  being  regarded 
as  decimals)  is  the  increase  of  the  sine  for  1".     Similarly,  if  the 
arc  be  24°  20',  the  increase  of  the  sine  for  1"  is  465,  the  last 
two  places  being  decimals.     The  same  remarks  are  equally 


12  DESCRIPTION   OF 

applicable  in  respect  of  the  column  D,  after  the  column  cosine, 
and  of  the  column  D,  between  the  tangents  and  cotangents. 
The  column  D,  between  the  tangents  and  cotangents,  is  equally 
applicable  to  either  of  these  columns ;  since  of  the  same  arc, 
the  log.  tang. + log.  cotang.=20  (24).  Therefore,  having  two 
arcs,  a  and  b,  log.  tang.  Z>-f-log.  cotang.  5= log.  tang,  a +log. 
cotang.  a;  or,  log.  tang,  b — log.  tang.  a=log.  cotang.  b — log. 
cotang.  a. 

26.  Now,  if  it  were  required  to  find  the  logarithmic  sine  of 
an  arc  expressed  in  degrees,  minutes,  and  seconds,  we  have 
only  to  find  the  degrees  and  minutes  as  before ;  then  multiply 
the  corresponding  tabular  number  by  the  seconds,  cut  off  two 
places  to  the  right-hand  for  decimals,  and  then  add  the  product 
to  the  number  first  found,  for  the  sine  of  the  given  arc.  Thus, 
if  we  wish  the  sine  of  40°  26'  28". 

The  sine  40°  26' 9.811952 

Tabular  difference  =  247 

Number  of  seconds  =     28 


Product  =  69.16,  which  being  added  =       69.16 


Gives  for  the  sine  of  40°  26'  28"  =  9.812021.16 

The  tangent  of  an  arc,  in  which  there  are  seconds,  is  found 
in  a  manner  entirely  similar.  In  regard  to  the  cosine  and 
cotangent,  it  must  be  remembered,  that  they  increase  while  the 
arcs  decrease,  and  decrease  while  the  arcs  are  increased ;  con- 
sequently, the  proportional  numbers  found  for  the  seconds  must 
be  subtracted,  not  added. 

Ex.  To  find  the  cosine  3°  40'  40". 

Cosine  3°  40' 9.999110 

Tabular  difference  =  13 

Number  of  seconds  =  40 

Product  =  5.20,  which  being  subtracted   =   5.80 


Gives  for  the  cosine  of  3°  40'  40"  =  9-999104.20 
27.   To  find  the  degrees,  minutes,  and  seconds  answering  to 
any  given  logarithmic  sine,  cosine,  tangent,  or  cotangent. 


THE    TABLES*  13 

Search  in  the  table,  and  in  the  proper  column,  until  the 
number  be  found ;  the  degrees  are  shown  either  at  the  top  or 
bottom  of  the  page,  and  the  minutes  in  the  side  columns,  either 
at  the  left  or  right.  But  if  the  number  cannot  be  exactly  found 
in  the  table,  take  the  degrees  and  minutes  answering  to  the 
nearest  less  logarithm,  the  logarithm  itself,  and  also  the  corres- 
ponding tabular  difference.  Subtract  the  logarithm  taken  from 
the  table  from  the  <riven  logarithm,  annex  two  ciphers,  and  then 
divide  the  remainder  by  the  tabular  difference  :  the  quotient  is 
seconds,  and  is  to  be  connected  with  the  degrees  and  minutes 
before  found ;  to  be  added  for  the  sine  and  tangent,  and  sub- 
tracted for  the  cosine  and  cotangent. 

Ex.  1.  To  find  the  arc  answering  to  the  sine  9.880054 
Sine  49°  20',  next  less  in  the  table,  9.879963 


Tab.  Diff.  181)9100(50" 

Hence  to  the  given  sign  9.880054  corresponds  the  arc  49° 

20' 50".  i 

Ex.  2.  To  find  the  arc  corresponding  to  cotang.  10.008688. 

The  given  cotang.  10.008688 
Cotang.  44°  26',  next  less  in  the  table,  10.008591 


Tab.  Diff.  421)9700(23" 

Hence,  44°  26'— 23" =44°  25'  37"  is  the  arc  corresponding 
to  the  given  cotangent  10.008688. 


OF  THE  TRAVERSE  TABLE. 

8.  A  table,  called  a  Traverse  Table,  is  used  in  computing 
the  area  of  a  survey  made  with  the  compass.  Its  use  will  be 
here  explained. 

This  table  shows  the  difference  of  latitude  and  departure, 
corresponding  to  any  bearing ;  and  for  any  distance  less  than 
100,  the  one  hundred  being  regarded  as  links,  chains,  rods,  or 
any  other  dimension. 

In  the  table  headed  « Traverse  Table,'  the  figures  at  the  top 
and  bottom  show  the  bearings,  to  degrees  and  parts  of  a  degree ; 
and  the  columns  on  the  left  and  right  of  each  page,  the  distances 
to  which  the  latitudes  and  departures  correspond. 


14  DESCRIPTION  OF 

If  the  bearing  be  less  than  45°,  the  angle  will  be  found  at 
the  top  of  the  page,  if  greater,  at  the  bottom ;  then,  if  the  dis- 
tance be  less  than  ,50,  it  will  be  found  in  the  columns  ("  dis- 
tances") of  the  left-hand  page ;  if  greater  than  50,  in  the 
columns  of  the  right-hand  page.  The  table  is  calculated  only 
to  quarter  degrees,  for  the  bearings  cannot  be  accurately  ascer- 
tained to  smaller  fractions  of  a  degree. 

29.  For  the  same  bearing,  and  lines  of  different  lengths,  it 
is  evident,  that  the  latitudes  and  departures  will  be  proportional 
'to  the  distances. 

Therefore,  when  the  distance  is  greater  than  100,  it  may  be 
divided  by  any  number  which  will  give  a  quotient  less  than 
100;  then,  the  latitude  and  departure  of  the  quotient  being 
multiplied  by  the  divisor,  the  products  are  the  latitude  and 
departure  of  the  whole  course.  It  is  also  plain,  that,  for  the 
same  bearing,  the  latitude  and  departure  of  the  sum  of  two 
or  more  distances,  is  equal  to  the  sum  of  the  latitudes  and 
departures  of  those  distances  respectively. 

Hence,  if  we  have  any  number  greater  than  100,  as  614,  we 
have  only  to  regard  the  last  figure  as  a  cipher,  and  recollect, 
that  610+4  =  614,  and  that  the  latitude  and  departure  of  610 
are  ten  times  as  great,  respectively,  as  the  latitude  and  departure 
of  61,  that  is,  equal  to  the  latitude  and  departure  of  61,  multi- 
plied by  10,  or,  with  the  decimal  point  removed  one  place  to 
the  right. 

Example  1.  To  find  the  latitude  and  departure,  the  bearing 
being  29  J°,  and  the  distance  582. 


Latitude  for  580         506.00 
Latitude  for      5  4.36 

Latitude  for  585         510.36 


Departure  for  580      283.40 
Departure  for       5          2.44 

Departure  for  585       285.84 


In  this  example,  the  latitude  and  departure  answering  to  the 
course  29  J°,  and  to  the  distance  58,  were  first  taken  from  the 
table,  and  the  decimal  point  moved  one  place  to  the  right; 
then  the  latitude  and  departure  answering  to  the  same  course, 
and  the  distance  5,  were  taken  from  the  table  and  added. 

Example  2.  To  find  the  latitude  and  departure,  the  bearing 
oeing  62^°,  and  the  distance  7855  chains. 


I 


THE    TABLES. 


15 


Latitude  7800 
Latitude       55 

Latitude  7855 

3602.00 
25.40 

3627.40 

Departure  7800 
Departure      5.0 

Departure  7855 

6919.00 
48.79 

6967.79 

If  the  distances  were  expressed  in  whole  numbers  and  deci- 
mals, the  manner  of  finding  the  latitudes  and  departures  would 
still  be  the  same,  except  in  pointing  off  the  decimal  places ; 
which,  however,  is  not  difficult,  when  it  is  remembered,  that  the 
column  of  distances  in  the  table  may  be  regarded  as  decimals 
by  removing  the  decimal  point  to  the  left  in  the  other  columns. 


*  * 


A   TABLE 


OP 


LOGARITHMS    OF    XUMBERS 

FROM     1     TO     10,000. 


N. 

Lg 

N. 

Log. 

N. 

Log. 

N. 

Lo$. 

1 
2 
3 
4 
5 

0.000000 
0.301030 
0.477121 
0.602060 
0.698970 

26 
27 
28 
29 
30 

1.414973 
*  1.431364 
1.447158 
1.462398 
1.477121 

51 
52 
53 
54 
55 

1  .  707570 
1.716003 
1  .  724276 
1  .  732394 
1  .  740363 

76 

77 
78 
79 
80 

1.880814 
.886491 
.892095 
.897627 
.903090 

6 
7 
8 
9 
10 

0.778151 
0.845098 
0.903090 
0.954243 
1.000000 

31 
32 
33 
34 
35 

1.491362 
1.505150 
1.518514 
1.531479 
1.544068 

56 
57 
58 
59 
60 

1.748188 
1.755875 
1.763428 
1  .  770852 
1.778151 

81 
82 
83 
84 
85 

.908485 
.913814 
.919078 
.924279 
.929419 

li 
12 
13 
14 
15 

1.041393 
1.079181 
1  .  1  13943 
1.146128 
1.176091 

36 
37 
38 
39 

40 

1.556303 
1.568202 
1.579784 
1.591065 
1.602060 

61 
62 
63 
64 
65 

1  .  785330 
1.792392 
1.799341 
1.806180 
1.812913 

86 
87 
88 
89 
90 

.934498 
.939519 
.  944483 
.949390 
.  954243 

16 
17 
18 
19 
20 

1.204120 
1.230449 
1.255273 
1.278754 
1.301030 

41 
42 
43 
44 
45 

1.612784 
1.623249 
1.633468 
1  .  643453 
1.653213 

66 
67 
68 
69 
70 

1.819544 
1.826075 
1.832509 
1.838849 
1.845098 

91 
92 
93 
94 
95 

.959041 
.963788 
.968483 
.973128 
.977724 

21 
22 
23 

24 
25 

1.322219 
1.342423 

1.361728 
1.380211 
1.397940 

46 
47 
48 
49 
50 

1.662758 
1.672098 
1.681241 
1.690196 
1.698970 

71 
72 
73 
74 
75 

1.851258 
1.857333 
1.863323 
1.869232 
1.875061 

96 
97 

98. 
99 
100 

.982271 
.986772 
.991226 
.995635 
2.000000 

N.B.  In  the  following  table,  in  the  last  nine  columns  of  each 
page,  where  the  first  or  leading  figures  change  from  9's  to  O's, 
points  or  dots  are  introduced  instead  of  the  O's  through  the  rest 
of  the  line,  to  catch  the  eye,  and  to  indicate  that  from  thence 
the  annexed  first  two  figures  of  the  Logarithm  in  the  second 
column  stand  in  the  next  lower  line. 


A  TABLE  OP  LOGARITHM  FROM  1  TO  10,000. 


N. 

Q     1  1  2    3  j  4    5  j  6  |  7 

8  |  9  |  D. 

100 

000000 

0434 

0868 

1301 

1734 

2J66 

2598 

3029 

3461 

3891 

432 

101 

4321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

102 

8600 

9026 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

424 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

419 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.361 

.775 

416 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

106 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

107 

9384 

9789 

,195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

404 

108 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

396 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4540 

4932 

393 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

389 

112 

.  9218 

9606 

9993 

.380 

.766 

1153 

1538 

1924 

2309 

2694 

386 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

382 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.320 

379 

115 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

376 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

372 

117 

8186 

8557 

8928 

9298 

9668 

..38 

.407 

.776 

1145 

1514 

369 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

366 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

9543 

9904 

.266 

.626 

.987. 

1347 

1707 

2067 

2426 

360 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

123 

9905 

.258 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

351 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5518, 

5866 

6215 

6562 

349 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

..26 

346 

126 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

340 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.253 

338 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

113943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

131 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

.245 

330 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

323 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

321 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

315 

138 

9879 

.194 

.508 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

314 

139 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

146128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

309 

141 

9219 

9527 

9835 

.142 

.449 

.756 

1063 

1370 

1676 

1982 

307 

142 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

301 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

299 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

151 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.985 

1272 

1558 

287 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

153 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

..51 

281 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

156 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

157 

5899 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

158 

8657 

8932 

9206 

9481 

9755 

..29 

.303 

.577 

.850 

1124 

274 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

N.    0|l 

2  |  3  I  4    5 

6  |  7 

8  |  9  |  D. 

A  TABLE  OF  LOGARITHMS  FROM  1  TO  10,000. 


•MM 

N. 

•••^••MMI 

0 

-^—  — 

1 

2 

3  |  4    5 

—  ^___ 

6 

—  ^—  —  - 

7 

"U    9  |  D. 

IfiO 

204120  4391 

4663  4934;  5204  5475 

5746  1  6016 

6286  6556 

271 

161 

6826 

7096 

7365  7634;  7904 

8173 

8441  8710 

8979 

9247 

269 

162 

9515 

9783|  ..51:  .319J  .586  .853 

1121 

1388 

1654 

1921 

267 

163 

212188 

2454  2720  ;  2986,  3252  3518  3783 

4049 

4314 

4579 

266 

164 
165 

4844 
7484 

5109i  5373  563815902 
7747  8010  8273!  8536 

6166  6430 
8798  9060 

6694J  6957 
9323  i  9585 

7221 
9846 

2o4 
262 

166 

220108 

0370  0631  1  0892  1153 

1414 

1675 

1936)2196 

2456 

261 

167 

2716 

2976:  3236  3496  3755  4015  4274 

4533 

4792 

5051 

259 

168 

6309 

5568  5826  6084  6342  6600  6858 

7115 

7372 

7630 

258 

169 

7887 

8144  8400i  8657  8913 

9170  9426 

9682 

9938 

.193 

256 

170 

230449 

0704  0960  j  1215  1470 

1724 

1979 

2234 

2488 

2742 

254 

171 

2996 

3250  3504J3757  4011 

4264 

4517 

4770 

5023 

5276 

253 

172 

5528 

5781  6033|6285  6537 

6789 

7041 

7292;  7544 

7795 

252 

173 

8046 

8297i  8548j  8799  i  9049 

9299 

9550 

9800  I  ..50 

.300 

250 

174 

240549 

0799  1048!  1297;  1546 

1795 

2044 

2293 

2541 

2790 

249 

175 

3038 

3286  3534  i  3782  4030 

4277 

4525 

4772 

5019 

5266 

248 

176 

5513 

5759  6006 

6252)  6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

7973 

82J9 

8464 

87091  8954 

9198 

9443 

9687 

9932 

.176 

245 

178 

250420  0664 

0908 

1151]  1395 

1638 

1881 

2125 

2368 

2610 

243 

179 

2853|  3096  3338  3580J  3822|  4064 

4306 

4548 

4790 

5031 

242 

180 

255273 

5514J  5755  59961  6237|  6477 

6718 

6958 

7198i  74391  241 

181 

7679 

7918  8158,  8393  8637  8877  9116 

93.55  95941  9833 

239 

182 

260071 

0310  0548  07871  1025  1263 

1501 

1739 

1976J  2214 

238 

183 

2451 

2883  ^92.5 

3162  3399  3636 

3873 

4109 

43461  4582 

237 

184 

4818  5054  5290 

5525 

5761 

5996 

6232 

6467!  6702 

6937 

235 

185 

7172  7406|  7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

186 

9513!  97461  9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

233 

187 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

189 

6462 

6692 

6921  7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754  8982 

9211 

9439 

9667 

9895 

.123 

.35l|  .578 

.806 

228 

191 

2810331  1261 

1488 

1715 

1942 

'2169 

2396 

2622 

2849 

3075 

227 

192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882  5107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681  6905 

7130 

7354 

7578 

225 

194 

7802 

802b 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

18T13 

2034 

222 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

197 

4466  4687 

4907 

5127 

5347  5567 

5787 

6007  6226 

6446 

220 

198 

6665  6884 

7104 

7323 

7542 

7761 

7979 

8198  8416 

8635 

219 

199 

88531  9071 

9289 

9507 

9725 

9943 

.161 

.378!  .595 

.813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

201 

3196 

3412  3628 

3844 

4059 

4275, 

4491 

4706 

4921 

5136 

216 

202 

5351 

5566  5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

203 

7496 

7710  7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

204 

9630  9843 

..56 

.268 

.481 

.693 

.906 

1118 

1330 

1542 

212 

205 

311754  1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

206 

3867  4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

207 

5970  6180  6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

208 

8063  i  8272;  8481 

8689  8898 

9106  9314 

9522 

9730 

993*8 

208 

209 

320146|  0354  0562 

0769  0977  1184  1391 

1598J  1805 

2012 

207 

210 

322219  2426 

2633  2839  3046  3252  3458  3665 

3871 

4077 

206 

211 

4282!  4488  4694  4899!  5105;  5310  5516  5721 

5926 

6131 

205 

212 

6336  6541  6745 

6950  i  7155  7359  j  7563  7767 

7972 

8176 

204 

213 

8380  85831  8787 

8991  9191  9398:  9601 

9805 

...8 

.211 

203 

214 

330414 

0617  0819 

1022  12251  I427t  1630  1832 

2034 

2236 

202 

215 

2438 

2640  2842 

3044  3248  3447;  3649;  3850  4051 

4253[  202 

216 

4454  46551  4856  5057;  5257  5458 

5658  5859 

6059 

6260 

201 

217 

6460  6660  6860  7060=  7260  •  7459 

7659  7858|  8058 

8257 

200 

218 

8456;  8656  88551  9054  9253'  9451 

9650  9849  1  ..47 

.246 

199 

219  340444  0642  OS411  10391  1237  1435  1632'  18301  2028  2225  198 

N.    0   !  1  I  2  I  3  1  4  |  5 

6    7 

8    9  i  D. 

A  TABLE  OF  LOGARITHMS  FROM  1  TO  10,000. 


N. 

0     1    2 

3(4    5|6|7|8|9 

D. 

220 

342423  2620 

2817 

3014  3212 

3409 

3606  3802 

3999 

4196 

197 

221 

4392  4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

222 

6353  6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

223 

8305  8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

..54 

194 

224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

193 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

229 

9835 

..25 

.215 

.404 

.593 

.783 

.972 

1161 

1350 

1539 

189 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

188 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

234 

9216 

9401 

9587 

9772 

9958 

.143 

.328 

.513 

.698 

.883 

185 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

237 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

..30 

181 

240 

380211 

0392 

0573 

0754 

0934 

11151  1296 

1476 

1656 

1837 

181 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636|  180 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428  !  179 

243 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

245 

9166 

9343 

9520 

9698 

9875 

..51 

.228 

.405 

.582 

.759 

177 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

176 

247 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

251 

9674 

9847 

..20 

.192 

.365 

.538 

.711 

.883 

1056 

1228 

173 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

3121 

3292 

3464 

3635 

3807 

3978 

414S 

4320 

4492 

4663 

171 

254 

4834 

5005 

5176 

5346 

55.7 

5688 

5858 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

169 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

414973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

263 

9956 

.121 

.286 

.451 

.616 

.781 

.945 

1110 

1275 

1439 

165 

264 

421604 

17G8 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

266 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

163 

267 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

162 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

9752 

9914 

..75 

.236 

.398 

.559 

.720 

.881 

1042 

1203 

161 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159 

273 

6163 

6322 

6481 

6640 

6798 

6957 

7116 

7275 

7433 

7592 

159 

274 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

275 

9333 

9491 

9648 

9806 

9964 

.122 

.279 

.437 

.594 

.752 

158 

276 

440909 

1066 

1224 

13811  1538 

1695 

1852 

2009 

2166 

2323 

157 

277 

2480 

^637 

2793 

2950 

3106  3263 

3419 

3576 

3732 

3889 

157 

278 

4045 

4201 

4357 

4513 

4669  4825 

4981 

5137 

5293 

5449 

156 

279 

5604  5760 

5915 

6071  62261  6382'  6537^  6692 

6848  1  7003 

155 

N. 

0 

1  1  2 

3  |  4  |  5  |  6  |  7  |  8 

9   D. 

A  TABLE  OF  LOGARITHMS  FROM   1   TO   10,000. 


N.  |   0     1    2    3    4|5J6|?l8 

9  |  D. 

280 

4471581  7313;  7468 

76231  7778 

7933 

8088 

8242 

8397 

8552 

155 

281 

8706|  8861 

9015 

9170  9324 

9478 

9633 

9787 

9941 

..95 

154 

282 
283 

450249 
1786 

0403 
1940 

0557 
2093 

0711  0865  1018 
2247!  2400J  2553 

1172 
2706 

1326 
2859 

1479 
3012 

1633 
3165 

154 
153 

284 

3318 

3471 

3624 

37771  3930  j  4082 

4235 

4387 

4540  4692 

153 

285 
286 

4845 
6366 

4997 
6518 

5150 
6670 

5302 
6821 

54o4 
6973 

5606 
7125 

5758 
7276 

5910 

7428 

6062  6214 
7579  7731 

152 
152 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

288 

9392 

9543 

9694 

9845 

9995 

.146 

.296 

.447 

.597 

.748 

151 

289 

46089S 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

290 

462398 

2548 

2697 

2847 

2997 

3146 

3296 

34451  3594 

3744 

150 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936  1  5085 

5234 

149 

292 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423  6571  j  67  19 

149 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

295 

9822 

9969 

.116 

.263 

.410 

.557 

.704 

.851 

.998 

1145 

147 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235!  5381 

5526 

146 

299 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300  I  477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

145 

301    8566 

8711 

8855 

899 

9143 

9287 

9431 

95751  9719 

9863 

144 

302 

480007 

0151 

0294 

0438 

0582 

0725 

08691  1012J  1156 

1299 

144 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302  1  2445  2588 

2731 

143 

304 

2874 

3016 

3159 

3302 

3445 

3587 

37301  3872  4015 

4157 

143 

305 

4300 

4442 

4585 

4727 

4869 

501  l|5153j  5295  5437 

5579 

142 

306 

5721 

5863 

6005 

6147 

6289 

6430!  6572!  6714<  6855 

6997 

142 

307 

7138 

7280  7421 

7563 

7704 

7845  7986 

8127 

8269 

8410 

141 

308 

8551 

8692 

8833 

8974 

9114 

9255i  9396 

9537 

9677 

9818 

141 

309 

9958 

..99 

.239 

.380 

.520 

.661 

.801 

.941 

1081 

1222 

140 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

311 

2760 

2900 

3040 

3179 

3319  3458  3597 

3737 

3876 

4015 

139 

312 

4155 

4294 

4433 

4572 

471114850  49891  5128J  5267 

5406 

139 

313 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

65151  6653 

6791 

139 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

315 

8311 

8448 

8586 

8724 

8862 

8999 

.9137 

9275 

9412 

9550 

138 

316 

9687 

9824 

9962 

..99 

.236 

.374 

.511 

.648 

.785 

.922 

137 

317 

501059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

318 

2427 

2564 

2700 

2837 

2973 

3109 

3246  i  3382 

3518 

3655 

136 

319 

3791 

3927 

4063 

4199 

432f5 

4471 

4607'  4743 

4878 

5014 

136 

320 

505150 

5286 

5421 

5557 

5693  5828 

5964!  6099 

6234 

6370 

136 

321 

6505 

6640 

6776 

6911 

7046  7181 

7316!  7451 

7586 

7721 

135 

322 

7856 

7991 

8126 

8260 

8395!  8530 

8664  8799 

8934 

9068 

135 

323 

9203 

9337 

9471 

9606 

9740 

9874 

...9  .143  .277 

.411 

134 

3-24 

510545 

0679 

0813 

0947 

1081 

1215 

1349  1482 

1616 

1750 

134 

325 

1883 

2017 

2151 

2284 

2418 

2551 

2684!  2818 

2951 

3084 

133 

326 

3218 

3351 

3484 

3617 

3750 

3883 

4016J  4149 

4282 

4414 

133 

327 

4548 

4681 

4S13 

4946 

5079 

5211 

5344i  5476 

5609 

5741 

133 

32S 

5874 

6006 

6139 

6271 

6403 

6535 

6668  6800 

6932 

7064 

132 

329 

7196 

7328 

7460 

7592  7724 

7855 

7987|8119i  8251 

8382 

132 

330 

518514 

8646 

8777 

8909 

9040 

9171 

9303 

9434 

9566 

9697 

131 

331 

9828 

9959 

..90 

.221 

.353 

.484 

.615 

.745 

.876 

1007 

131 

332 

521138 

1269 

1400 

1530 

1661 

1792 

1922i  2053 

2183 

2314 

131 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226  3356 

3486 

3616 

130 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4526i  4656 

4785 

4915 

130 

335 

5045 

5174 

5304 

5434 

5563 

5693 

5822  j  5951 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114!  7243 

7372 

7501 

129 

337 

7630 

7759 

7888 

80161  8145 

8274 

8402^  8531 

8660 

8788 

129 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687  9815 

9943 

..72 

128 

339 

530200 

0328>  0456 

0584 

0712 

0840 

0968'  1096 

1223 

1351 

128 

N.  1   0     1  1  2 

3    4 

5  |  6    7 

8 

9   D. 

A  TABLE  OF  LOGARITHMS  FKOM  1  TO  10,000. 


N. 

0 

1 

2  |  3    4  |  5 

6  |  7  |  8  |  9  |  D. 

340 

531479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2500 

2627 

128 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3999 

127 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

344 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

..79 

.204 

125 

347 

540329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

125 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

38ro 

3944 

124 

350 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

124 

351 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635!  8758 

8881 

123 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.106 

123 

355 

550228 

0351 

0473 

6595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

359 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

556303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

9907 

..26 

.146 

.265 

.385 

.504 

.624 

.743 

.863 

.982 

119 

364 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

366 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

367 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

568202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

117 

371 

9374 

9491 

9608 

9725 

9842 

9959 

..76 

.19b 

.309 

.426 

117 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

1476 

1592 

117 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

375 

.4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

9898 

..12 

.126 

.241 

.355 

.469 

.583 

.697 

.811 

114 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

385 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

388 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112 

389 

9950 

..61 

.173 

.284 

.396 

.507 

.619 

.730 

.842 

.953 

112 

59l) 

591065 

1176 

1287 

1399 

1510 

1G21 

1732 

1843 

1955 

2066 

111 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

397 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

9556 

9665 

9774 

109 

398 

9883 

9992 

.101 

.210 

.319 

.428 

.537 

.646 

.  755 

.864 

109 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

N. 

0 

1    2  |  3  I  4 

5  i  6  |  7  |  8  !  9  I  D. 

A  TABLE  OF  LOGARITHMS  FROM    1    TO    10,000. 


N.    0     1  1  2    3    4    5  |  6    7  !  8 

9 

D 

400  602060 

2169 

2277 

2386  2494;  2H031 

2711 

2819 

2928  3036'  108 

401 

3144 

3253 

3361 

3469  3577  3686 

3794 

3902 

4010 

41181  108 

402 

4226 

4334 

4442 

4550  465S  4766 

4874 

4982 

5089 

5197 

108 

403 

5305 

5413 

5521 

5628  5736 

5844 

5«51 

6059 

6166 

6274 

108 

404 

6381 

6489 

6596 

6704|  68  11 

6919 

7026 

7133 

7241 

7348 

107 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

107 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274  9381 

9488 

107 

407 

9594'  9701 

9808 

9914 

..21 

.128 

.234  .341 

.447 

.5541  107 

408 
409 

61066010767 
1723|  1829 

0873 
1936 

0979J  1086 
20421  2148 

1192 
2254 

1298 
2360 

1405 
2466 

1511 
2572 

1617 

2678 

106 
106 

410 

612/84 

28901  2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

106 

411 

3842 

39471  4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

412 

4897 

5003  5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

413 

5950 

60551  6160 

6265 

6370 

6476 

6581 

6686  6790 

6895 

105 

414 

7000 

7105!  7210 

7315 

7420 

7525 

7629 

7734  7839!  7943 

105 

415 

8048 

8153  8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

416 

9093 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

..32 

104 

417 

620136 

0240 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

101 

418 

1176 

1280 

1384|  1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

419)   2214 

2318 

2421  2525 

2628 

2732 

2835 

2939 

3042 

3146 

ir>4 

420  I  623249 

3353 

3456  3559 

3663^  3766 

3869 

3973  4076 

4179 

t03 

421    4282 

4385 

4488  4591 

4695 

4798 

4901 

5004 

5107!  5210 

103 

422 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135  6238 

103 

423 

6340 

64-43 

6546 

6648 

6751 

6853 

6956  7058  1  7  16  lj  7263  103 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082;  8185 

8287J  102" 

425 

8389 

8491 

8598 

8695 

8797 

8900 

9002 

9104!  9206 

93081  102 

426 

9410 

9512 

9613 

9715 

9817 

9919 

..211  .123;  .224 

.326J  102 

127 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139!  1241 

1342 

102 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255" 

235G 

101 

4l,!9 

2457 

•2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266  3367 

101 

480 

633468 

J569 

3670 

3771 

3872 

3973 

4074 

4175  4276 

4376 

100 

431 

4477 

4578 

4679 

4779 

4880  4981 

5081 

51821  5283 

5383 

100 

432 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

433 
434 

6488 
7490 

6588 
7590 

6688 
7690 

6789 
7790 

6889 
7890 

6989 
7990 

7089 
8090 

7189 
8190 

7290 
8290 

7390!  100 
8389i  99 

435 

8489 

8589 

8689 

8789 

8888  8988 

9088 

9188 

9287 

9387 

99 

436 

9486 

9586 

9686 

9785 

9885  9984 

..84 

.183 

.283 

.382 

99 

437 

640481 

0581 

0680  j  0779 

0879  0978 

1077 

1177 

1276 

1375 

99 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267  2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

643453 

3551 

3650  3749 

3847 

3946 

4044J  4143 

4242 

4340 

98 

441 

4439 

4537 

4636 

4734  4832 

4931 

5029 

5127 

5226 

5324 

98 

442 

5422 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

98 

443 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

444 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

445 

8360 

8458 

8555  8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

446 

9335 

9432 

9530!  9627 

9724 

9821 

9919 

..16 

.113 

.210 

97 

447 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

97 

44S 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

449 

2246 

2343 

2440J  2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

450 

653213 

3309 

3405  3502 

3598!  3695 

3791 

3888  3984 

4080 

96 

451 

4177 

4273 

4369!  4465 

4562  4658 

4754 

48501  4946 

5042 

96 

452 

5138 

5235 

533  li  5427 

5523  5619 

5715 

5810|  5906 

6002 

96 

453 

6098 

6194 

6290  6386 

6482J  6577 

6673 

6769 

6864 

6960 

96 

454 

7056 

7152 

7247 

7343 

7438i  7534 

7629 

7725 

7820 

7916 

96 

455 

8011 

8107 

B203 

8298 

8393  j  8488 

8584 

8679 

8774 

8870 

95 

456 

8965 

9060 

9155  9250 

9346  9441 

9536 

9631 

9726 

9821 

95 

457 

9916 

..11 

.1061  .201 

.296 

.391 

.486 

.581 

.676 

.771 

95 

458 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

95 

459 

HI  3 

1907 

2002 

2096 

2191 

228C 

2380 

2475 

2569 

2663 

95 

N.  1   0 

1234 

5    6 

7    8    9   D. 

A  TABLE  OF  LOGAKITHMS  FROM  1   TO  10,000. 


N.  o  ;  i 

2  |  3 

4    5 

6  |  7  |  8    9  |  D. 

460 

662758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

3607 

94 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

462 

4642 

4736 

4830 

4924 

5018 

5112 

5200 

5299 

5393 

5487 

94 

463 

5581 

5675 

5769 

§862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

464 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

465 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

467 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

..60 

.153 

93 

468 

670241 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

93 

469 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

92 

471 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

92 

472 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

473 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

474 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

475 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

476 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

477 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

478 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

..63 

.154 

.245 

91 

479 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

91 

480 

681241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

90 

481 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

482 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

485 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

486 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

489 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

.,19 

.107 

89 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

0905 

0993 

89 

491 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

88 

492 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

88 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

494 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

495 

4605 

4693 

4781 

4868 

4956 

5044 

5i31 

5219 

5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

87 

501 

9838 

9924 

..11 

..98 

.184 

.271 

.358 

.444 

.531 

.617 

87 

502 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

86 

503 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

504 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

86 

505 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3895 

3979 

4065 

86 

506 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

507 

5008 

5094 

5179 

5265  5350 

5436 

5522 

5607 

5693 

5778 

86 

508 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

6718 

6803 

6888 

6974 

7059 

7144 

2229 

7315 

7400 

7485 

85 

510 

707570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

85 

511 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

85 

512 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

..33 

85 

513 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

85 

514 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

84 

515 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

84 

516 

265C 

2734 

2815 

2902 

298fi 

3070 

3154 

3238 

3323 

3407 

84 

517 

349 

3575 

365C 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

518 

433C 

4414 

4497 

4581 

4665 

474S 

4833 

4916 

5000 

5084 

84 

519 

516- 

5251 

533£ 

5418 

55021  558C 

5669 

5753 

583P 

5920 

84 

N. 

0   I  1 

2  |  3  !  4  |  5    6  |  7  |  8  |  9  |  D. 

A  TABLE  OF  LOGARITHMS  FROM   1   TO   10,000. 


N.  |   0     1    2    3    4    5    fi  |  7    8  |  9  |  D. 

520 

716003 

6087;  6170  6254 

6337 

6421 

6504  H588  6671 

6754  83 

521 
522 

6838 
7671 

6921 
7754 

1  7004 
7837 

7088 
7920 

7171 
8003 

7254 
8086 

7338  7421  7504 
8169  8253  8336 

7587 
8419 

83 
83 

523 

8502 

8585 

8668 

8751 

8834J  8917 

9000  9083:9165 

9248 

83 

524 

9331 

9414 

9497 

9580 

96631  9745 

9828  9911  9994 

..77 

83 

525 

720159 

0242 

0325 

0407 

0490!  0573 

0655 

0738  0821 

0903 

83 

526 

0986 

1068 

1151 

1233 

1316)  1398 

1481 

1563  1646 

1728 

82 

527 

1811 

1893 

1975 

2058 

2140  2222 

2305  2387  2469 

2552 

82 

528 

2634 

2716|  2798 

2881 

2963  3045 

3127|  3209,  3291 

3374 

82 

529 

3456 

3538!  3620 

3702 

3784  3866 

3948 

4030 

4112 

4194 

82 

530 

724276 

4358  4440 

4522 

4604  4685 

4767 

4849 

4931 

5013 

82 

531 

5095 

5176  5258 

5340 

5422  5503 

5585 

5667 

5748 

5830 

82 

532 

5912 

5993  6075 

6156 

6238  6320 

6401 

6483 

6564 

6646 

82 

533 

6727 

68091  6890 

6972 

7053)  7134 

7216 

7297 

7379 

7460 

81 

534 

7541 

76231  7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

535 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

537 

9974 

..55 

.136 

.217 

.298 

.378 

.459 

•540 

.621 

.702 

81 

538 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

539 

1589 

1C69 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

732394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

80 

541 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320  j  4400 

4480 

4560 

4640 

4720 

80 

543 

4800 

4880 

4960 

5040 

5120|  5200 

5279 

5359 

5439 

5519 

80 

544 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

80 

545 

6397 

6476  6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

546 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

549 

9572 

9651 

9731 

9810 

9889 

9968 

..47 

.126 

.205 

.284 

79 

550 

740363 

0442 

0521 

0600 

0678 

0757 

0836 

0915 

0994 

1073 

79 

551 

1152 

1230 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2646 

79 

553 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

554 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

78 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

559 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748188 

8266 

8343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

77 

561 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

562 

9736 

9814 

9891 

9968 

..45 

.123 

.200 

.277 

.354 

.431 

77 

563 

750508 

0586  0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

1279 

1356  1433 

1510 

1587 

1664 

1741 

1818 

1895 

1972 

77 

565 

2048 

2125  2202 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

566 

2816 

2893|  2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

567 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

568 

4348 

4425 

4501  4578 

4654 

4730 

4807 

4883 

4960 

5036 

76 

569 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

755875 

5951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

76 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

572 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

8003 

8079 

76 

573 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761  8836 

76 

574 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

575 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.196 

.272 

.347 

75 

576 

760422 

0498 

0573 

0648 

0724 

0799 

0875 

0950 

10251  1101 

75 

577 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778|  1853 

75 

578 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529J  2604 

75 

579 

2679 

27541  2829 

2904'  *v<8l3053 

3128 

3203 

3278'  3353 

75 

N.  |   0   |  1    2 

3  |  4    5 

6  |  7 

8  1  9   D. 

A  TABLE  OF  LOGARITHMS  FKOM  1  TO  10,000. 


N. 

0   I   1 

2 

3  |  4  |  5  |  6 

7 

8 

9  !  D. 

580 

76342S 

3503 

357fc 

3653 

3727 

3802  3877 

3952 

4027 

4101 

75 

581 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

75 

582 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

583 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

584 

6413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

585 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

587 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

588 

9377 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

..42 

74 

589 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

74 

590 

770852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

"74 

591 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

592 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

594 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

595 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

596 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

73 

597 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778151 

8224 

8296 

8368J  8441 

8513 

8585 

8658 

8730 

8802 

72 

601 

8874 

8947 

9019 

9091  19163 

9236 

9308 

9380 

9452 

9524 

72 

602 

9596 

9669 

9741 

9813 

9885 

9957 

..29 

.101 

.173 

.245  72 

603 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

604 

1037 

1109 

'1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2136 

2258 

2329 

2401 

72 

606 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

607 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

~71 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

612 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

71 

613 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

71 

614 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

615 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

616 

9581 

9651 

9722 

9792 

9863 

9933 

...4 

..74 

.144 

.215 

70 

617 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

618 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

619 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

622 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

624 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

>5672 

5741 

5811 

70 

625 

5880 

5949 

6ul9 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

626 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

627 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890 

69 

628 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

629 

8651 

8720 

8789 

8858 

8927 

8996 

9065  9134 

9203 

9272 

69 

630 

799341 

9409 

9478 

9547 

9616 

9685 

9754  9823 

9892 

9961 

69 

631 

800029 

0098 

0167 

0236 

0305 

0373 

04421  0511 

0580 

06481  69 

632 

0717 

0786 

0854 

0923 

0992  1061 

1129 

1198 

1266 

1335 

69 

633 

1404 

1472 

1541 

1609 

1678  1747 

1815 

1884 

1952 

2021 

69 

634 

2089 

2158 

2226 

2295 

2363!  2432 

2500 

2568 

2637 

2705 

69 

635 

2774 

2842 

2910 

2979. 

3047  3116  3184 

3252 

3321 

3389 

68 

636 

3457 

3525- 

3594 

3662 

3730  3798 

3867 

3935 

4003 

4071 

68 

637 

4139 

4208 

4276 

4344 

44121  4480 

4548 

4616 

4685 

4753 

68 

638 

4821 

4889 

4957 

5025 

50931  5161 

5229 

52<J7 

5365 

5433 

68 

639 

5501 

5569 

5637 

5705 

57731  5841 

5908 

59^6 

6044 

6112'  68 

N. 

0 

i   |2|3|4|5|6|7|8|9JD. 

A  TABLE  OF  LOGARITHMS  FKOM  I  TO  10,000. 


11 


N. 

0   I  1 

2  I  3  [  4 

5  |  6  |  7 

8  |  9 

D. 

640 

1T06180I  6248 

6316J  6384 

6451 

6519 

6587 

6655 

6723 

6791) 

68 

641 

6858 

6926 

6994|  7061 

7129 

7197 

7264 

7332 

7400 

7467 

68 

642 

7535 

7603 

7670]  7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

643 

8211 

8279 

8346  8414 

8481 

8549 

8616 

8684 

8751 

8818 

67 

644 

8886 

8953 

9021 

9088 

9156 

9223 

9290j  9358 

9425 

9492 

67 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964  ..31 

.,98 

.165 

67 

646 

810233 

0300 

0367 

0434 

0501 

0569 

0636 

0703 

0770 

0837 

67 

647 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

648 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

649 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

67 

651 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

652 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5046 

5113  5179 

5246 

5312 

5378 

5445 

5511 

66 

654 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

655 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

656 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

658 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

9610 

9676 

9741 

9807 

9873 

9939 

...4 

..70 

.136 

66 

661 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

66 

662 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

663 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

65 

665 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

666 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

65 

667 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

668 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

669 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

65 

670 

826075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

672 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

673 

8015 

8080 

8144 

82091  8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9141 

9175 

9239 

64 

675 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

64 

676 

•  9947 

..11 

..75 

.139 

.204 

.268 

.332 

•396 

.460 

.525 

64 

677 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

678 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

2573 

2637 

2700 

2764  2828 

2892 

2956 

3020 

3083 

64 

681 

3147 

3211 

3275 

3338 

3402  |  3466 

3530 

3593 

3657 

3721 

64 

682 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

683 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

5056 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

685 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

686 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

688 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

689 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

63 

691 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918!  9981 

..43 

63 

692 

840106 

0169  0232 

0294 

0357 

0420 

0482 

0545]  0608 

0671 

63 

693 

0733 

0796  0859 

0921 

0984 

1046 

1109 

11721  1234 

1297 

63 

694 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797  1860 

1922 

63 

695 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422  i  2484]  2547 

62 

696 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046  3108 

3170 

62 

697 

3233 

3295 

3357 

3420 

3482 

3544j 

.3606 

36691  3731 

3793 

62 

698 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

429  li  4353 

4415 

62 

699 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912!  4974  5036 

62 

N.  |   0   |  1 

2  |  3 

4  |  5  |  6  |  7 

8 

9  |  D. 

12 


A  TABLE  OP  LOGARITHMS  FROM   1  TO  10,000 


N. 

0  -  |  1  |  2  |  3    4 

5 

6  |  7 

8 

9  |  D. 

700 

845098 

5160.  5222 

5284 

5346 

5408 

5470 

5532 

5594 

569& 

62 

701 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

702 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

703 

6955 

7017 

7079 

7141 

7202 

7264 

7320 

7388 

7449 

7511 

62 

704 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

705 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

706 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

707 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

61 

709 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

851258 

1320 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

61 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

713 
714 

3090 
3698 

3150 
3759 

3211 
3820 

3272 

3881 

3333 
3941 

3394 
4002 

3455 
40',  3 

3516  3577 
412414185 

3637 
4245 

61 
61 

715 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

717 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

718 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

719 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

60 

721 

7935 

799t5 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

•  722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

724   9739 

9799 

9859 

9918 

9978 

..38 

..98 

.158 

.218 

.278 

60 

725 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

1  1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

728 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

60 

729 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

•  60 

730 

863323' 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

59 

731 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

733 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

734 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

735 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

736 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

69 

737 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

738 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

59 

741 

9818 

9877 

9935 

9994 

..53 

.111 

.170 

.228 

.287 

.345 

59 

742 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

743 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

744 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

745 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

746 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

58 

747 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

749 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

875061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

58 

751 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

755 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

758 

9669 

9726 

9784 

9841 

9898 

9956 

..13 

..70 

.127 

.185 

57 

759 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

57 

N. 

0 

1  1  2  |  3 

4 

5 

6  |  7 

8  |  9  !  D. 

A  TABLF  OF  LOGARITHMS  FEOM   1   TO   10,000. 


N.    0|l    2|3|4    5    6    7|8 

9  |  D. 

700 

880814 

0871 

0928 

0985  1042 

1099 

1156 

1213 

1271 

1328 

57 

761 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

57 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

763 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

764 

3093 

3150 

3207 

32641  3321 

3377 

3434 

3491 

3548  3605 

57 

-7H5 

3661 

3718 

3775 

3832  3888 

3945 

4002 

4059 

4115  4172 

57 

766 

4229 

4285 

4342 

4399  4455 

4512 

4569 

4625 

4682  4739 

57 

767 

4795 

4852 

4909 

4965  5022 

5078 

5135 

5192 

5248  5305 

57 

768 

5361 

5418 

5474 

5531  5587 

5644 

5700 

5757 

5813 

5870 

57 

769 

5926 

5983 

6039 

6096 

6152 

6209  6265 

6321 

6378 

6434 

56 

770 

886491 

6547 

6604 

6660 

6716 

6773J  6829 

6885 

6942 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

7336  j  7392 

7449  7505 

7561 

56 

772 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011  8067 

8123 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8516  8573 

8629 

8685 

56 

774 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

56 

775 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9750 

9806 

56 

776 

9862 

9918 

9974 

..30 

..86 

.141 

.197 

.253 

.309 

.365 

56 

777 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

T78 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

779 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

56 

781 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039  4094 

4150 

4205 

4261 

55 

784 

4316 

4371 

4427 

4482 

4538 

4593  1  4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

4980 

5036 

5091 

5146  5201 

5257 

5312 

5367 

55 

786 

5423 

5478 

5533 

5588 

5644 

5699  5754 

5809 

5864 

5920 

55 

787 

5975 

6030 

6085 

6140 

6195 

6251  6306 

6361 

6416 

6471 

55 

788 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897627 

7682 

7737 

7792 

7847 

7992 

7957 

8012 

8067 

8122 

55 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

794 

9821 

9875 

9930 

9985 

..39 

..94 

.149 

.203 

.258 

.312 

55 

795 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

79G 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

797 

1458 

1513 

1567 

1622 

1676 

1731!  1785 

'1840 

1894 

1948 

54 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

800 

903090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

54 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

804 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

805 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

808 

7411 

7465 

7519 

7573 

7626 

7680!  7734 

7787 

7841 

7895 

54 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

54 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

812 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

..37 

53 

813 

910091 

0144 

0197 

0251 

0304 

0358  10411  0464 

0518 

0571 

53 

814 

0624 

0678 

0731 

0784 

0838  089  li  0944!  0998 

1051 

1104 

53 

815 

1158 

1211 

1264 

1317 

1371 

1424|  1477 

1530 

1584 

1637 

53 

816 

1690 

1743 

1797 

1850 

1903 

1956  2009 

2063 

2116 

2169 

53 

817 

2222 

2275 

2328 

2381 

2435 

2488  2541  2594 

2647 

2700 

53 

818 

2753 

2806 

2859 

2913 

2966  3019  3072  3125 

3178 

3231 

53 

819 

3284 

3337 

3390 

3443 

3496  3549  3602  3655 

3708 

3761 

53 

N.    0|l 

2 

3 

4  |  5 

6 

7  |  8  |  9 

D. 

14 


A  TABLE  OP  LOGARITHMS  FROM  1  TO  10,000. 


N.  |   0   |   1  !  2 

3  1  4  |  5 

6  |  7    8 

9  |  D. 

820 

913814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

42901  53 

821 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

822 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

53 

823 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

53 

824 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

828 

8030 

8083 

8135  8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

831 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

..19 

..71 

52 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

52 

833 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

836 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985  3037 

3089 

3140 

3192 

52 

838 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

52 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

4331 

43*3 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

52 

841 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

52 

842 

5312 

5364  5415 

5467 

5518 

5570'  5621 

5673 

5725 

5776 

52 

843 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

844 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

51 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

849 

8908 

8959 

9010  9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

850 

929419 

9470 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

51 

851 

9930 

9981 

..32 

..83 

.134 

.185 

.236 

.287 

.338 

.389 

51 

852 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

51 

853 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

51 

854 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

51 

860 

934498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

50 

861 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

50 

862 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

864 

.6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

50 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

04J7 

0467 

50 

872 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

874 

1511 

1561 

1611 

J660 

1710 

1760 

1809 

1859 

1909 

1958 

50 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

876 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433'  49 

N.    0|l    2  |  8    4    5    6|7|8'9|D. 

A  TABLE  OF  LOGARITHMS  FROM   1  TO  10,000. 


15 


N. 

0 

1  1  2 

3  |  4    5    6  |  7  |  8  |  9 

D 

&41 

944433 

4532 

4581 

4631 

4tiS() 

47291  4779 

4828 

4S77 

4927 

49 

881 

4976 

5025 

5074 

5124 

5173 

5222  5272 

5321 

5370 

5419 

49 

S-.JO 

5469 

5518 

5567 

5616 

5665 

5715  5764 

5813 

5862 

5912 

49 

>S3 

5961 

6010 

6059 

6108 

6157 

62071  6256 

6305 

6354 

6403 

49 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

885 

6943 

69921  7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

886 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

888 

8413 

8463 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

8902  8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49 

890 

949390 

9439 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

49 

891 

9878 

9926 

9975 

..24 

..73 

.121 

.170 

.219 

.267 

.316 

49 

892 

950365 

0414 

0462 

0511 

0560 

0608 

OG57 

0706 

0754 

0803 

49 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

12*9 

49 

894 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49 

8<*5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

896 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696  1  2744 

48 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48 

899 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

4291 

4339 

4387  4435 

4484 

4532 

4580  4628 

4677 

48 

901 

4725 

4773 

4821 

4869  4918 

•i960 

5014 

5062  5110 

5158i  48 

902 

5207 

5255 

5303 

5351  5399 

5147 

5495 

5543  5592 

56401  48 

903 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024  6072  6120  48 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505  6553  6601  48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

69841  7032 

7080 

48 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464!  7512 

7559 

48 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

KM 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

85]  6!  48 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

88981  8946 

8994  48 

910 

959041 

9089 

9137 

9185 

9232 

9280 

9328 

9375  9423 

9471  48 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947  48 

912 

9995 

..42 

..90 

.138 

.185 

.233 

.280 

.328 

.376 

.423!  48 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279]  1326 

1374 

47 

915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753  1801 

1848 

47 

916 

1895 

1943 

1990 

2038 

2085)2132 

2180 

2227  2275 

2322 

47 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

270l|  2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

919 

3316 

3363 

3410 

3457 

3504 

3552 

3r99 

3646 

3693 

3741 

47 

920 

963788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108  5155 

47 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

55781  56251  47 

924 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095  47 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470  6517 

6564^  47 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

69391  6986 

7033 

47 

927 

708a 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

47 

928 

7548 

7595 

7642 

7688 

77301  7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

8109 

8156 

8203!  8249 

8296 

8343 

8390 

8436:  47 

930 

9168483 

8530 

8576 

8623 

8670  8716 

8763 

8810 

8856 

8903i  47 

931 

8950 

8996 

9043 

9090 

9136!  9183 

9229 

9276 

9323 

9369  47 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835  47 

933 

9882 

9928  9975 

..21 

..68 

.114 

.161 

.207 

.254  .3001  47 

934 

970347 

0393 

0440 

0486 

0533  0579 

0626 

0*72 

0719  1  07651  46 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

113? 

1183 

1229  46 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693|  46 

937 

174C 

1786 

1833 

1879J  1925 

1971 

2018 

2064 

t  2110 

21571  46 

938 

2203 

2249  2295 

2342,2388  2434 

2481 

25271  2573 

2619'  46 

939 

266f 

27121  2758!  2804  2851  2897:  2943 

2389'3035  3082  46 

N. 

0 

1  1  2 

3  |  4    5    6  |-  7  |  8 

9 

D. 

16 


A  TABLE  OP  LOGARITHMS  FROM  1   TO  10,000. 


N. 

0   I  1 

2|3|4|5|6|7|8|9|D. 

940 

973128 

317413220 

3266 

33131  3359 

3405 

3451  ( 

3497 

3543 

46 

941 

3590 

3636  3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

46 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4S34 

4880 

4926 

46 

944 

4972 

5018 

5064 

5110 

5106 

5202 

5248 

5294 

5340 

5386 

46 

945 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

948 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

949 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

950 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

46 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

45 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

45 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

962 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

963 

3626 

3671 

3716 

3762 

3807 

3852  3897 

3942 

3987 

4032 

45 

964 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

966 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

45 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

45 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.250 

.294 

44 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

44 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

44 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

44 

989 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

995635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

44 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6U68 

44 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

997 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

44 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

N.  |   0   |  1    2    3  I  4  |  5  1  6  |  7 

8  |  9   D. 

4  TABLE 


OF 

LOGARITHMIC 
SINES   AND    TANGENTS, 

FOR  EVERY 

DEGREE    AND    MINUTE 

OP   THE    QUADRANT. 


N.B.  The  minutes  in  the  left-hand  column  of  each  page, 
increasing  downwards,  belong  to  the  degrees  at  the  top  ;  and 
those  increasing  upwards,  in  the  right-hand  column,  belong  to 
the  degrees  below. 


18 


(0  Degree.)     A  TABLE  OF  LOGARITHMIC 


M. 

Sine   |   D. 

Cosine    D.  |   Tang.     D.      Cotang.  | 

0 

0.000000 

10.000000 

O.OOOOOOl 

Ininme. 

bO 

1 

6.463726 

501  T17 

000000 

00 

6.463726 

501717 

13.536274 

59 

2 

764756 

293485 

000000 

00 

764756 

293483 

235244 

58 

3 

940847 

208231 

000000 

00 

940847 

208231 

059i53 

57 

4 

7.065786 

161517 

000000 

00 

7.065786 

161517 

12.934214 

56 

5 

162696 

131968 

000000 

00 

162696 

131969 

837304 

55 

6 

241877 

111575 

9.999999 

01 

241878 

1115-78 

758122 

54 

7 

308824 

96653 

999999 

01 

308825 

99653 

691175 

53 

8 

366816 

85254 

999999 

01 

366817 

85254 

633183 

52 

9 

417968 

76263 

999999 

01 

417970 

76263 

582030 

51 

10 

463725 

68988 

999998 

01 

463727 

68988 

5362731  50 

11 

7.505118 

62981 

9.999998 

01 

7.505120 

62981 

12.494880 

49 

12 

542906 

57936 

999997 

01 

542909 

57933 

457091 

48 

13 

577668 

53641 

999997 

01 

577672 

53642 

422328 

47 

14 

609853 

49938 

999996 

01 

609857 

49939 

390143 

46 

15 

639816 

46714 

999996 

01 

639820 

46715 

360180 

45 

16 

667845 

43881 

999995 

01 

667849 

43882 

332151 

44 

17 

694173 

41372 

999995 

01 

694179 

41373 

305821 

43 

18 

7]8997 

39135 

999994 

01 

719003 

39136 

280997 

42 

19 

742477 

37127 

999993 

01 

742484 

37128 

257516 

41 

20 

764754 

35315 

999993 

01 

764761 

35136 

235239 

40 

21 

7.785943 

33672 

9.999992 

01 

7.785951 

33673 

12.214049 

39 

22 

806146 

32175 

999991 

01 

806155 

32176 

193845 

38 

23 

825451 

30805 

999990 

01 

825460 

30806 

174540 

37 

24 

843934 

29547 

999989 

0-2 

843944 

29549 

156056 

36 

25 

861662 

28388 

999988 

02 

861674 

28390 

138326 

35 

26 

878695 

27317 

999988 

0-2 

878708 

27318 

121292 

34 

27 

895085 

26323 

999987 

0'2 

895099 

26325 

104901 

33 

28 

910879 

25399 

999986 

02 

910894 

25401 

089106 

32 

29 

926119 

24538 

999985 

02 

926134 

24540 

073866 

31 

30 

940842 

23733 

999983 

02 

940858 

23735 

059142 

30 

31 

7.955082 

22980 

9'.  999982 

0-2 

7.955100 

22981 

12.044900 

29 

32 

968870 

22273 

999981 

0-2 

968889 

22275 

031111 

28 

33 

982233 

21608 

999980 

02 

982253 

21610 

017747 

27 

S4 

995198 

20981 

999979 

02 

995219 

2J983 

004781 

26 

35 

8.007787 

20390 

999977 

02 

8.007809 

2J392 

11.992191 

25 

36 

020021 

19831 

999976 

02 

020045 

19833 

979955 

24 

37 

031919 

19302 

999975 

02 

031945 

19305 

968055 

23 

38 

043501 

18801 

999973 

02 

043527 

18803 

956473 

22 

39 

054781 

18325 

999972 

02 

054809 

18327 

945191 

21 

40 

065776 

17872 

999971 

02 

065806 

17874 

934194 

20 

41 

8.076500 

17441 

9.999969 

02 

8.076531 

17444 

11.923469 

19 

42 

086965 

17031 

999968 

02 

086997 

17034 

913003 

18 

43 

097183 

16639 

999966 

02 

097217 

16642 

902783 

17 

44 

107167 

16265 

999964 

03 

107202 

16268 

892797 

16 

45 

116926 

15908 

999963 

03 

116963 

15910 

883037 

15 

46 

126471 

15566 

999961 

03 

126510 

15568 

873490 

14 

47 

135810 

15238 

999959 

03 

135851 

15241 

864149 

13 

48 

144953 

14924 

999958 

03 

144996 

14927 

855004 

12 

49 

153907 

14622 

999956 

03 

153952 

14627 

846048 

11 

SJ 

163681 

14333 

999954 

03 

162727 

14336 

837273 

10 

51 

8.171280 

14054 

9.999952 

03 

8.171328 

14057 

11.828672 

9 

52 

179713 

13786 

999950 

03 

179763 

13790 

820237 

8 

53 

187985 

13529 

999948 

1)3 

188036 

13532 

811964 

7 

54 

196102 

13280 

999946 

03 

196156 

13284 

803844 

6 

55 

204070 

13041 

999944 

;j 

204126 

13044 

795874 

5 

56 

211895 

12810 

999942 

4 

211953 

12814 

788047 

4 

57 

219581 

12587 

999940 

04 

219641 

12590 

780359 

3 

58 

227134 

12372 

999938 

04 

227195 

12376 

772805 

2 

59 

234557 

12164 

999936 

04 

234621 

12168 

765379 

1 

60 

241855 

11963 

999934 

04 

241921 

11967 

758079 

0 

Cosine 

Sine    1   j   Cotang.         J    Tang. 

M. 

Degrees. 


SINES    AND    TANGENTS.       (1    DeglCC.) 


M.  |   Sine   |   D.     Cosine    D.   Tang;. 

D.     Cotang.   | 

0 

8.241855 

11963 

9.999934 

04l  8.241921 

11967 

1  1  .  758079 

60 

1 

249033 

11768 

999932 

04 

249102 

11772 

750898 

59 

2 

256094 

1  1580 

999929 

04 

256165 

11584 

743835 

58 

3 

26304^ 

11398 

999927 

01 

263115 

11402 

736885 

57 

4 

269881 

11221 

999925 

04 

269956 

11225 

730044 

56 

5 

276614 

11050 

999922 

04 

276691 

11054 

723309 

55 

6 

283243 

10883 

999920 

04 

283323 

10887 

716677 

54 

7 

289773 

10721 

999918 

04 

289856 

10726 

710144 

53 

8 

296207 

10565 

999915 

M 

296292 

10570 

703708 

52 

9 

302546 

10413 

999913 

04 

302634 

10418 

697366 

51 

10 

308794 

10266 

999910 

04 

308884 

10270 

691116 

50 

11 

8.314954 

10122 

9.999907 

04 

8.315046 

10126 

11.684954 

49 

12 

321027 

9982 

999905 

04 

321122 

9987 

678878 

48 

13 

327016 

9847 

999902 

04 

327114 

9851 

672886 

47 

14 

332924 

9714 

999899 

05 

331025 

9719 

666975 

46 

15 

338753 

9586 

999897 

Go 

33SS56'  9590 

661144 

45 

16 

344504 

9460 

999894 

05 

344610 

9465 

655390 

44 

17 

350181 

9338 

999891 

05 

350289 

9343 

649711 

43 

18 

355783 

9219 

999888 

06 

355895 

9224 

644105 

42 

19 

361315 

9103 

999885 

05 

361430 

9108 

638570 

41 

20 

366777 

8990 

999882 

05 

366895 

8995 

633105 

40 

21 

8.372171 

8880 

9.999879 

05 

8.372292 

8885 

11.627708 

39 

22 

377499 

8772 

999876 

05 

377622 

8777 

622378 

38 

23 

382762 

8667 

999873 

06 

382889 

8672 

617111 

37 

24 

387962 

8564 

999870 

05 

388092 

8570 

611908 

36 

25 

393101 

8464 

999867 

05 

393234 

8470 

606766 

35 

26 

398179 

8366 

999864 

05 

398315 

8371 

601685 

34 

27 

403199 

8271 

999861 

05 

403338 

8276 

596662 

33 

28 

408161 

8177 

999858 

05 

408304 

8182 

591696 

32 

29 

413068 

8086 

999854 

05 

413213 

8091 

586787 

31 

30 

417919 

7996 

999851 

06 

418068 

8002 

581932 

30 

31 

8.422717 

7909 

9.999848 

06 

8.422869 

7914 

11.577131 

29 

32 

427462 

7823 

999844 

06 

427618 

7830 

572382 

28 

33 

432156 

7740 

999841 

06 

432315 

7745 

567685 

27 

34 

436800 

7657 

999838 

06 

436962 

7663 

563038 

26 

35 

441394 

7577 

999834 

06 

441560 

7583 

558440 

25 

36 

445941 

7499 

999831 

06 

446110 

7505 

553890 

24 

37 

450440 

7422 

999827 

06 

450613 

7428 

549387 

23 

38 

454893 

7346 

999823 

06 

455070 

7352 

544930 

22 

39 

459301 

7273 

999820 

06 

459481 

7279 

540519 

21 

40 

463665 

7200 

999816 

06 

463849 

7206 

536151 

20 

41 

8.467985 

7129 

9.999812 

06 

8.468172 

7135 

11.531828 

19 

42 

472263 

7060 

999809 

06 

472454 

7066 

527546 

18 

43 

476498 

6991 

999805 

06 

476693 

6998 

523307 

17 

44 

480693 

6924 

999801 

06 

480892 

6931 

519108 

16 

45 

484848 

6859 

999797 

07 

485050 

6865 

514950 

15 

46 

488963 

6794 

999793 

07 

489170 

6801 

510830 

14 

47 

493040 

6731 

999790 

07 

493250 

6738 

506750 

13 

48 

497078 

6669 

999788 

07 

497293 

6676 

502707 

12 

49 

501080 

6608 

999782 

07 

501298 

6615 

498702 

11 

50 

505045 

6548 

999778 

07 

505267 

6555 

494733 

10 

51 

8.508974 

6489 

9.999774 

07 

8,509200 

6496 

11.490800 

9 

52 

512867 

6431 

999769 

07 

513098 

6439 

486902 

8 

53 

516726 

6375 

999765 

07 

516961 

6382 

483039 

7 

54 

520551 

6319 

999761(07 

5207P9 

6326 

479210 

6 

55 

524343!  6264 

999757  07 

5245k  5 

6272 

475414 

5 

56 

528102 

6211 

999753 

07 

528349 

6218 

471651 

4 

57 

531828 

6158 

999748 

or 

532080 

6165 

467920 

3 

58 

535523 

6106 

999744 

or 

535779 

6113 

464221 

2 

59 

539186 

6055 

999740 

or 

539447 

6062 

460553 

1 

60 

542819 

6004    99973ft 

or 

543084 

6012 

456916 

0 

Cosine 

Sine 

Cotang. 

Tang    !  Al. 

db  Degrees 


20 


(2  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M. 

Sine   |   D.     Cosine   |  I). 

Tanp.   i   D. 

Cnnuv. 

0 

8.542819 

6004 

9.999735 

07 

8.543084 

6012 

1  1.4569ibi  00 

1 

546422 

5955 

999731 

07 

54669  1 

5962 

453309 

59 

2 

549995 

5906 

999726 

07 

550268 

5914 

449732 

58 

3 

553539 

5858 

999722 

08 

553817 

5866 

*  446183 

57 

4 

557054 

5811 

999717 

08 

557336 

5819 

442664  56 

5 

560540 

5765 

999713 

08 

560828 

5773 

439172  55 

6 

563999 

5719 

999708 

08 

564291 

5727 

4357091  54 

7 

567431 

5674 

999704 

08 

567727 

5682 

4322731  53 

8 

570836 

5630 

999699  08 

571137 

5638 

428863  52 

9 

574214 

5587 

999694 

08 

574520 

5595 

425480  51 

10 

577566 

5544 

999689 

08 

577877 

5552 

422123 

50 

11 

8.580892 

5502 

9.999085 

OS 

8.581208 

5510 

11.418792 

49 

12 

584193 

5460 

999680 

08 

584514 

5468 

415486  48 

13 

587469 

5419 

999675 

08 

587795 

5427 

412205|  47 

14 

590721 

5379 

9^9670 

08 

591051 

5387 

408949  46 

15 

593948 

5339 

999665 

08 

594283 

5347 

405717]  45 

*6 

597152 

5300 

999660 

08 

597492 

5308 

4025081  44 

17 

600332 

5261 

999655 

08 

600677 

5270 

3993231  43 

18 

603489 

5223 

999650 

08 

603839 

5232 

396161 

42 

19 

606623 

5186 

999645 

09 

606978 

5194 

393022 

41 

20 

609734 

5149 

999640 

<>9 

610094 

5158 

389906 

40 

21 

8.612823 

5112 

9.999035 

09 

8.613189 

5121 

11.386811 

39 

22 

615891 

5076 

999629 

09 

616262 

5085 

383738 

38 

23 

618937 

5041 

999624 

09 

619313 

5050 

380687 

37 

24 

621962 

5006 

999619 

09 

622343 

5015 

377657 

36 

25 

624965 

4972 

999614 

OH 

625352 

4981 

374648 

35 

26 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34 

27 

631)911 

4904 

999603 

09 

631308 

4913 

368692 

33 

28 

633854 

4871 

999597 

09 

634256 

4880 

365744 

32 

29 

636776 

4839 

999592 

09 

637184 

4848 

3628  J  6 

31 

30 

639680 

4806 

999586 

09 

640093 

4816 

359907 

30 

31 

8.642563 

4775 

9.999581 

09 

8  .  642982 

4784 

11.357018- 

29 

32 

645428 

4743 

999575 

09 

645853 

4753 

354147 

28 

33 

648274 

4712 

999570 

09 

648704 

4722 

351296 

27 

34 

651102 

4682 

999564 

09 

651537 

4691 

348463 

26 

35 

653911 

4652 

999558 

10 

"654352 

4661 

345648 

25 

36 

656702 

4622 

999553 

10 

657149 

4631 

34285  1 

24 

37 

659475 

4592 

999547 

10 

659928 

4602 

340072 

23 

38 

662230 

4563 

999541 

10 

662689 

4573 

337311 

22 

39 

664968 

4535 

999535 

10 

665433 

4544 

3345671  21 

40 

667689 

4506 

999529 

10 

668160 

4526 

331  840  j  20 

41 

8.670393 

4479* 

9.999524 

10 

8.670870 

4488 

11.329130 

19 

42 

673080 

4451 

999518 

10 

673563 

4461 

326437 

18 

43 

675751 

4424 

999512 

10 

676239 

4434 

323761 

17 

44 

678405 

4397 

999506 

10 

678900 

4417 

321100 

16 

45 

681043 

4370 

999500 

10 

681544 

4380 

318456 

15 

46 

683665 

4344 

999493 

10 

684172 

4354 

315828 

14 

47 

686272 

4318 

999487 

10 

686784 

4328 

313216 

13 

48 

688863 

4292 

999481 

10 

689381 

4303 

310619 

12 

49 

691438 

4267 

999475 

10 

691963 

4277 

308037 

11 

50 

693998 

%  4242 

999469 

10 

694529 

4252 

305471 

10 

51 

8.696543 

4217 

9.999463 

11 

8.697081 

4228 

11.302919 

9 

52 

699073 

4192 

999456 

11 

699617 

4203 

300383 

8 

53 

701589 

4168 

999450 

11 

702139 

4179 

297861 

7 

54 

704090 

4144 

999443 

11 

704646 

4155 

295354 

6 

55 

706577 

4121 

999437 

11 

707140 

4132 

292860 

5 

56 

709049 

4097 

999431 

11 

709618 

4108 

290382 

4 

57 

711507 

4074 

999424 

11 

712083 

4085 

287917 

3 

58 

713952 

4051 

999418 

11 

714534 

4062 

285465 

2 

59 

716383 

4029 

999411 

11 

716972 

4040 

283028 

1 

60 

718800 

4006 

999404 

11 

719396 

4017 

280304 

0 

Cosine   j       |   Sine   I 

Cotanc.  1 

T3nc. 

M. 

H7  Degrees. 


SINES  AND  TANGENTS.     (3  Degrees.) 


21 


M. 

Sine   | 

D. 

Cosine 

n. 

Tang. 

D. 

CotHne. 

0 

S.  718800 

4006 

9.9994U4 

ll 

8.719396 

4017 

11.280604 

60 

1 

721204 

3984 

999398 

11 

721806 

3995 

278194 

59 

2 

723595 

3962 

999391 

11 

724204 

3974 

275796 

58 

3 

725972 

3941 

999384 

11 

726588 

3952 

273412 

57 

4 

728337 

3919 

999378 

ll 

728959 

39JO 

271041 

56 

5 

730688 

3898 

999371 

ll 

731317 

3909 

268683 

55 

6 

733027 

3877 

999364 

12 

733663 

3889 

266337 

54 

7 

735354 

3857 

999357 

I  '2 

735996 

3868 

264004 

53 

8 

737607 

3836 

999350 

12 

738317 

3848 

261683 

52 

9 

739969 

3816 

999343 

12 

740626 

3827 

259374 

51 

10 

742259 

3796 

999336 

I) 

742922 

3807 

257078 

50 

11 

8.744536 

3776 

9.999329 

12 

8.745207 

3787 

11.254793 

49 

12 

746802 

3756 

999322 

12 

747479 

3768 

252521 

48 

13 

749055 

3707 

999315 

ia 

749740 

3749 

250260 

47 

14 

751297 

3717 

999308 

12 

751989 

3729 

248011 

46 

15 

753528 

3698 

999301 

IS 

754227 

3710 

245773 

45 

16 

755747 

3679 

999294 

14 

756453 

3692 

243547 

44 

17 

757955 

3661 

999286 

12 

758668 

3673 

241332 

43 

18 

760151 

3642 

999279 

12 

760872 

3655 

239128 

42 

19 

762337 

3624 

999272 

12 

763065 

3636 

236935 

41 

20 

764511 

3606 

999265 

12 

765246 

3618 

234754 

40 

21 

8.766675 

3588 

9.999257 

12 

8.767417 

3600 

11.232583 

39 

22 

768828 

3570 

999250 

13 

769578 

3583 

230422 

38 

23 

770970 

3553 

999242 

13 

771727 

3565 

228273 

37 

24 

773101 

3535 

999235 

13 

773866 

3548 

226134 

36 

25 

775223 

3518 

999227 

13 

775995 

3531 

224005 

35 

26 

777333 

3501 

999220 

13 

778114 

3514 

221886 

34 

27 

779434 

3484 

999212 

13 

780222 

3497 

219778 

33 

28 

781524 

3467 

999205 

j  ': 

782320 

3480 

217680 

32 

29 

*  783605 

3451 

999197 

13 

784408 

3464 

215592 

31 

30 

785675 

3431 

999189 

ia 

786486 

3447 

213514 

30 

31 

8.787736 

3418 

9.999181 

13 

8.788554 

3431 

11.211446 

29 

32 

789787 

3402 

999174 

ia 

790613 

3414 

209387 

28 

33 

791828 

3386 

999166 

13 

792662 

3399 

207338 

27 

34 

793859 

3370 

999158 

13 

794701 

3383 

205299 

26 

35 

795881 

3354 

999150 

13 

796731 

3368- 

203269 

25 

36 

797894 

3339 

999142 

13 

798752 

3352 

201248 

24 

37 

799897 

3323 

999134 

ia 

800763 

3337 

199237 

23 

38 

801892 

3308 

999126 

0 

ft 

802765 

3322 

197235 

22 

39 

803876 

3293 

999118 

• 

804758 

3307 

195242 

21 

40 

805852 

3278 

999110 

'. 

806742 

3292 

193258 

20 

41 

8.807819 

3263 

9.999102 

I 

8.808717 

3278 

11.191283 

19 

42 

809777 

3249 

999094 

/ 

810683 

3262 

189317 

18 

43 

811726 

3234 

999086 

4 

812641 

3248 

187359 

17 

44 

813667 

3219 

999077 

4 

814589 

3233 

185411 

16 

45 

815599 

3205 

999069 

.: 

816529 

3219 

183471 

15 

46 

817522 

3191 

999061 

.'. 

818461 

3205 

181539 

14 

47 

819436 

3177 

999053 

.'. 

820384 

3191 

179616 

13 

48 

821343 

3163 

999044 

{ 

822298 

3177 

177702 

12 

49 

823240 

3149 

999036 

,'. 

824205 

3163 

175795 

11 

50 

825130 

3135 

999027 

.: 

826103 

3150 

173897 

10 

51 

8.827011 

3122 

9.999019 

14 

8.827992 

3136 

11.172008 

9 

52 

828884 

3108 

999010 

14 

829874 

3123 

170126 

8 

53 

830749 

3095 

999002 

14 

831748 

3110 

168252 

7 

54 

832607 

3082 

998993 

14 

833613 

3096 

166387 

6 

55 

834456 

3069 

998984 

14 

835471 

3083 

164529 

5 

56 

836297 

3056 

998976 

14 

837321 

3070 

162679 

4 

57 

838130 

3043 

998967 

15 

839163 

3057 

160837 

3 

58 

839956 

3030 

998958 

15 

840998 

3045 

159002 

2 

59 

841774 

3017 

998950 

15 

842825 

3032 

157175 

1 

60 

843585 

3000 

998941 

15 

844644 

3019 

155356 

0 

Cosine 

Sine 

Cotang. 

Tang. 

M. 

86  Degrees. 


(4  Degrees.)    A  TABLE  OF  LOGARITHMIC 


M.    Sine 

D.     Cosine    D.  |   Tang.      D. 

Cotang. 

U 

8.843585 

3005 

9.998941 

15 

8.844644 

3019 

11.155356 

60 

1 

845387 

2992 

998932 

15 

846455 

3007 

153545 

59 

2 

847183 

2980 

998923 

15 

848260 

2995 

151740 

58 

3 

848971 

2P67 

998914 

15 

850057 

2982 

149943 

57 

4 

850751 

2955 

998905 

15 

851846 

2970 

148154 

56 

5 

852525 

2943 

998896 

15 

853628 

2958 

146372 

55 

6 

854291 

2931 

998887 

15 

855403 

2946 

144597 

54 

7 

856049 

2919 

998878 

15 

857171 

2935 

142829 

53 

8 

857801 

2907 

998869 

15 

858932 

2923 

141068 

52 

9 

859546 

2896 

998860 

15 

860686 

2911 

139314 

51 

10 

861283 

2884 

998851 

15 

862433 

2900 

137567 

50 

11 

.8.863014 

2873 

9.998841 

15 

8.864173 

2888 

11.135827 

49 

12 

864738 

2861 

998832 

15 

865906 

2877 

134094 

48 

13 

666455 

2850 

998823 

16 

867632 

2868 

132368 

47 

14 

868165 

2839 

998813 

16 

869351 

2854 

130649 

46 

15 

869868 

2828 

998804 

16 

871064 

2843 

128936 

45 

16 

871565 

2817 

998795 

16 

872770 

2832 

127230 

44 

17 

873255 

2806 

998785 

16 

874469 

2821 

125531 

43 

18 

874938 

2795 

998776 

16 

876162 

2811 

123838 

42 

19 

876615 

2786 

998766 

16 

877849 

2800 

122151 

41 

20 

878285 

2773 

998757 

16 

879529 

2789 

120471 

40 

21 

8.879949 

2763 

9.998747 

16 

8.881202 

2779 

11.118798 

39 

22 

881607 

2752 

998738 

16 

882869 

2768 

117131 

38 

23 

883258 

2742 

998728 

16 

884530 

2758 

115470 

37 

24 

884903 

2731 

998718 

16 

886185 

2747 

113815 

36 

25 

886542 

2721 

998708 

16 

887833 

2737 

112167 

35 

26 

888174 

2711 

998699 

16 

889476 

2727 

110524 

34 

27 

889801 

2700 

998689 

16 

891112 

2717 

108888 

33 

28 

891421 

2690 

998679 

16 

892742 

2707 

107258 

32 

29 

893035 

2680 

998669 

17 

894366 

2697 

105634 

31 

30 

894643 

2670 

998659 

17 

895984 

2687 

104016 

30 

31 

8.896246 

2660 

9.998649 

17 

8.897596 

2677 

11.102404 

29 

32 

897842 

2651 

998639 

17 

899203 

2667 

100797 

28 

33 

899432 

2641 

998629 

17 

900803 

2658 

099197 

27 

34 

901017 

2631 

998619 

17 

902398 

2648 

097602 

26 

35 

902596 

2622 

998609 

17 

903987 

2638 

096013 

25 

36 

904169 

2612 

998599 

17 

905570 

2629 

094430 

24 

37 

905736 

2603 

998589 

17 

907147 

2620 

092853 

23 

38 

907297 

2593 

998578 

17 

908719 

2610 

091281 

22 

39 

908853 

2584 

998568 

17 

910285 

2601 

089715 

21 

40 

910404 

2575 

998558 

17 

911846 

2592 

088154 

20 

41 

8.911949 

2566 

9.998548 

17 

8.913401 

2583 

11.086599 

19 

42 

913488 

2556 

998537 

17 

914951 

2574 

085049 

18 

43 

915022 

2547 

998527 

17 

916495 

2565 

083505 

17 

44 

916550 

2538 

.  998516 

18 

918034 

2556 

081966 

16 

45 

918073 

2529 

998506 

18 

919568 

2547 

080432 

15 

46 

919591 

2520 

998495 

18 

921096 

2538 

078904 

14 

47 

921103 

2512 

998485 

18 

922619 

2530 

077381 

13 

48 

922610 

2503 

998474 

18 

924136 

2521 

075864 

12 

49 

924112 

2494 

998464 

18 

925649 

2512 

074351 

11 

50 

925609 

2486 

998453 

18 

927156 

2503 

072844 

10 

51 

8.927100 

24^7 

9.998442 

18 

8.928658 

2495 

11.071342 

9 

52 

928587 

2469 

998431 

18 

930155 

2486 

069845 

8 

53 

930068 

2460 

998421 

18 

931647 

2478 

068353 

7 

54 

931544 

2452 

998410 

18 

933134 

2470 

066866 

6 

55 

933015 

2443 

998399 

18 

934616 

2461 

065384 

5 

56 

934481 

2435 

998388 

18 

936093 

2453 

063907 

4 

57 

935942 

2427 

998377 

18 

937565 

2445 

062435 

3 

58 

937398 

2419 

998366 

18 

939032 

2437 

060968 

2 

59 

938850 

2411 

998355 

18 

940494 

2430 

059506 

1 

60 

940296 

2403 

998344 

18 

941952 

2421  J 

058048 

0 

Cosine 

|   Sine   | 

Cutang. 

|   Tang.   |  M. 

85  Degrees. 


SINES  AND  TANGENTS.     (5  Degrees.) 


M.  |   Sine   |   D.     Cosine   |  D.  |   Tang.      D. 

Ootang. 

0 

8.940296 

2403 

9.99834* 

19 

8.941952 

2421 

11.058048 

60 

1 

941738 

2394 

998333 

19 

943404 

2413 

056596 

59 

2 

943174 

2387 

998322 

19 

944S52 

2405 

055148 

58 

3 

944606 

2379 

998311 

19 

946295 

2397 

053705 

57 

4 

946034 

2371 

998300 

19 

947734 

2390 

052266 

56 

tt 

947456 

2363 

998289 

19 

949168 

2382 

050832 

55 

f. 

948874 

2355 

998277 

19 

950597 

2374 

049403 

54 

V 

950287 

2348 

998266 

19 

952021 

2366 

047979 

53 

| 

951696 

2340 

998255 

19 

953441 

2360 

046559 

52 

C 

953100 

2332 

998243 

19 

954856 

2351 

045144 

51 

10 

954499 

2325 

998232 

19 

956267 

2344 

043733 

50 

11 

8  .  955894 

2317 

9.998220 

19 

8.957674 

2337 

11.042326149 

15 

957284 

2310 

998209 

19 

959075 

2329 

040925 

48 

IS 

958670 

2302 

998197 

19 

960473 

2323 

039527 

47 

14 

960052 

2295 

998186 

19 

961866 

2314 

038134 

46 

If- 

961429 

2288 

998174 

19 

963255 

2307 

036745 

45 

10 

962801 

2280 

998163 

19 

964639 

2300 

035361 

44 

17 

964170 

2273 

998151 

19 

966019 

2293 

033981 

43 

18 

965534 

2266 

998139 

20 

967394 

2286 

032606 

42 

Ifl 

966893 

2259 

998128 

20 

968766 

2279 

031234 

41 

20 

968249 

2252 

998116 

20 

970133 

2271 

029867 

40 

21 

8.969600 

2244 

9.998104 

20 

8.971496 

2265 

11.028504 

39 

22 

970947 

2238 

998092 

20 

972855 

2257 

027145 

38 

23 

972289 

2231 

998080 

20 

974209 

2251 

025791 

37 

24 

973628 

2224 

998068 

20 

975560 

2244 

024440 

36 

25 

974962 

2217 

998056 

20 

976906 

2237 

0230941  35 

26 

,  976293 

2210 

998044 

20 

978248 

2230 

021752 

34 

27 

977619 

2203 

998032 

20 

979586 

2223 

020414 

33 

28 

978941 

2197 

998020 

20 

980921 

2217 

019079 

32 

29 

980259 

2190 

998008 

20 

982251 

2210 

017749 

31 

30 

981573 

2183 

997996 

20 

983577 

2204 

016423 

30 

31 

8.982883 

2177 

9.997984 

20 

8.984899 

2197 

11.015101 

29 

32 

984189 

2170 

997972 

20 

986217 

2191 

013783 

28 

33 

985491 

2163 

997959 

20 

987532 

2184 

012468 

27 

34 

986789 

2157 

997947 

20 

988842 

2178 

011158 

26 

35 

988083 

2150 

997935 

21 

990149 

2171 

009851 

25 

36 

989374 

2144 

997922 

21 

991451 

2165 

008549 

24 

37 

990660 

2138 

997910 

21 

992750 

2158 

007250 

23 

:3^ 

991943 

2131 

997897 

21 

994045 

2152 

005955 

22 

39 

993222 

2125 

997885 

21 

995337 

2146 

004663 

21 

40 

994497 

2119 

997872 

21 

996624 

2140 

003376 

20 

41 

8.995768 

2112 

9.997860 

21 

8.997908 

2134 

11.002092 

19 

42 

997036 

2106 

997847 

21 

999188 

2127 

000812 

18 

43 

998299 

2100 

997835 

21 

9.000465 

2121 

10.999535 

17 

44 

999560 

2094 

997822 

21 

001738 

2115 

998262 

16 

45 

9.000816 

2087 

997809 

21 

003007 

2109 

996993 

15 

46 

002069 

2082 

997797 

21 

004272 

2103 

995728 

14 

47 

003318 

2076 

997784 

21 

005534 

2097 

994466 

13 

48 

004563 

2070 

997771 

21 

006792 

2091 

993208 

12 

49 

005805 

2064 

997758 

21 

008047 

2085 

991953 

11 

50 

007044 

2058 

997745 

21 

009298 

2080 

990702 

0 

51 

9.008278 

2052 

9.997732 

21 

9.010546 

2074 

10.989454 

9 

52 

009510 

2046 

997719 

21 

011790 

2068 

988210 

8 

53 

010737 

2040 

997706 

21 

013031 

2062 

986969 

7 

54 

011962 

2034 

997693 

22 

014268 

2056 

985732 

6 

55 

013182 

2029 

997680 

22 

015502 

2051 

984498 

5 

56 

014400 

2023 

997667 

22 

016732 

2045 

983268 

4 

57 

015613)  2017 

997654 

22 

017959 

2040 

982041 

3 

58 

0168241  201? 

907641 

22 

019183 

2033 

980817 

2 

59 

018031:  2006 

997628 

22 

020403 

2028 

979597 

1 

60 

019235  2000 

997614 

22 

021620 

2023 

978380 

0 

C»sine  | 

Sine   |   j  Cotang.   j^ 

Tang. 

M. 

84  Degrees. 


24 


(6  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.  |   Sine 

D.   |  Cosine   |  I).    Tang.  |   D. 

Cotang. 

0 

9.019235) 

2000 

9.997614 

22  9.021620 

2023 

0.978380 

60 

1 

0204351 

1995 

997601 

22 

022834 

2017 

977166 

59 

2 

021632 

1989 

997588 

22 

024044 

2011 

975956 

58 

3 

022825 

1984 

997574 

22 

025251 

2006 

974749 

57 

4 

024016 

1978 

997561 

22 

026455 

2000 

973545 

56 

5 

025203 

1973 

997547 

22 

027655 

1995 

972345 

55 

6 

026386 

1967 

997534 

23 

028852 

1990 

971148 

54 

7 

027567 

1962 

997520 

23 

030046 

1985 

969954 

53 

8 

028744 

1957 

997507 

23 

031237 

1979 

968763 

52 

9 

029918 

1951 

997493 

23 

032425 

1974 

£•67575 

51 

10 

031089 

1947 

997480 

23 

033609 

1969 

966391 

50 

11 

9.032257 

1941 

9.997466 

23 

9.034791 

1964 

10.965209 

49 

12 

033421 

1936 

997452 

23 

0359691  1958 

964031 

48 

13 

034582 

1930 

997439 

23 

037144 

1953 

962856 

47 

14 

035741 

1925 

997425 

23 

038316 

1948 

961684 

46 

15 

036896 

1920 

997411 

23 

039485 

1943 

960515 

45 

16 

038048 

1915 

997397 

23 

040651 

1938 

959349 

44 

17 

039197 

1910 

997383 

23 

041813 

1933 

958187 

43 

18 

040342 

1905 

997369 

23 

042973 

1928 

957027 

42 

19 

041485 

1899 

997355 

23 

044130 

1923 

955870 

41 

20 

042625 

1894 

997341 

23 

045284 

1918 

954716 

40 

21 

9.043762 

1889 

9.997327 

24 

9.046434 

1913 

10.953566 

39 

22 

044895 

1884 

997313 

24 

047582 

1908 

952418 

38 

23 

046026 

1879 

997299 

24 

048727 

1903 

951273 

37 

24 

047154 

1875 

997285 

24 

049869 

1898 

950131 

36 

25 

048279 

1870 

997271 

24 

051008 

1893 

948992 

35 

26 

049400 

1865 

997257 

24 

052144 

1889 

947856 

34 

27 

050519 

1860 

997242 

24 

053277 

1884 

946723 

33 

28 

051635 

1855 

997228 

24 

054407 

1879 

945593 

32 

29 

052749 

1850 

997214 

24 

055535 

1874 

944465 

31 

30 

053859 

1845 

997199 

24 

056659 

1870 

943341 

30 

31 

054966 

1841 

9.997185 

24 

9.057781 

1865 

10.942219 

29 

32 

0560711  1836 

997170 

24 

058900 

1869 

941100 

28 

33 

057172 

1831 

997156 

24 

060016 

1855 

939984 

27 

34 

058271 

1827 

997141 

24 

061130 

1851 

938870 

26 

35 

059367 

1822 

997127 

24 

062240 

1846 

937760 

25 

36 

060460 

1817 

997112 

24 

063348 

1842 

936652 

24 

37 

061551 

1813 

997098 

24 

064453 

1837 

935547 

23 

38 

062639!  1808 

997083 

25 

065556 

1833 

934444 

22 

39 

063724 

1  1804 

997068 

25 

066655 

1828 

933345 

21 

40 

064806 

1799 

997053 

25 

067752 

1824 

932248 

20 

41 

9.065885!  1794 

9.997039 

25  9.068846 

1819 

10.931154 

19 

42 

066962  1790 

997024 

25 

069938 

1815 

930062 

18 

43 

068036 

1786 

997009 

25 

071027 

1810 

928973 

17 

44 

069107  1781 

996994 

25 

072113 

1806 

927887 

16 

45 

070176!  1777 

996979 

25 

073197 

1802 

926803 

15 

46 

071242 

1772 

996964 

25 

074278 

1797 

925722 

14 

47 

072306|  1768 

996949 

25 

075356 

1793 

924644 

13 

48 

073366 

1763 

996934 

25 

'  076432 

1789 

923568 

12 

49 

074424  !  1759 

996919 

25 

077505 

1784 

922495 

11 

50 

075480 

1755 

996904 

25 

078576 

1780 

921424 

10 

51 

9.076533 

1750 

9.996889 

25 

9.079644 

1776 

10.920356 

9 

52 

*.  0775831  1746 

996874 

25 

080710 

1772 

919290 

8 

53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

7 

54 

079676 

1738 

996843 

25 

082833 

1763 

917167 

6 

55 

080719 

1733 

996828 

25 

083891 

1759 

916109 

c 

56 

081759 

1729 

996812 

26 

084947 

1755 

915053 

4 

57 

082797 

1725 

996797 

26 

086000 

1751 

914000 

£ 

58 

083832 

1721 

996782 

26 

087050  1747 

912950 

c 

59 

084864 

1717 

996766 

26 

08S098 

1743 

911902 

1 

60 

085894 

1713    996751 

26 

089144 

1738 

910856 

0 

Cosine 

I   Sine   1 

Coiang. 

Tang. 

M. 

83  Degrees. 


SIXES  AND  TANGENTS.     ^7  Degrees.) 


25 


M.  |    Sine      D.     Cosine   |  D.  |   Tang.   j   D. 

Cotang.   J 

0 

9.  085894 

1713 

9.996751'  26 

9.089144 

1738 

0.910850 

60 

1 

086922 

1709 

996735  26 

090187 

1734 

909813 

59 

2 

087947 

1704 

996720  26 

091228 

1730 

908772 

58 

3 

088970 

1700 

996704  26 

092266 

1727 

907734 

57 

4 

089990 

1696 

996688  26 

093302 

1722 

906698 

56 

5 

091008 

1692 

996673:  26 

094336 

1719 

905664 

55 

6 

092024 

1688 

996657  26 

095367 

1715 

9046^,3 

54 

7 

093037 

1684 

996641126 

096395 

1711 

903605 

53 

8 

094047 

1680 

996625  26 

097422 

1707 

902578 

52 

9 

095056 

1676 

996610  26 

098446 

1703 

901554 

51 

10 

096062 

1673 

996594  26 

099468 

1699 

900532 

50 

11 

9.097065 

1668 

9.996578  27 

9.100487 

1695 

10.899513 

49 

12 

098066 

1665 

996562  27 

101504 

1691 

898496 

48 

13 

099065 

1661 

996546  27 

102519 

1687 

897481 

47 

14 

100062 

1657 

996530 

27 

103532 

1684 

896468 

46 

15 

101056 

1653 

996514 

27 

104542 

1680 

895458 

45 

16 

102048 

1649 

996498 

27 

105550 

1676 

894450 

44 

17 

103037 

1645 

996482 

27 

106556 

1672 

893444 

43 

18 

104025 

1641 

996465 

27 

107559 

1669 

892441 

42 

19 

105010 

1638 

996449 

27 

108560 

1665 

891440 

41 

20 

105992 

1634 

996433 

27 

109559 

1661 

890441 

40 

21 

9.106973 

1630 

9.996417 

27 

9.110556 

1658 

10.889444 

39 

22 

107951 

1627 

996400 

27 

111551 

1654 

888449 

38 

23 

108927 

1623 

996384 

27 

112543 

1650 

887457 

37 

24 

109901 

1619 

996368 

27 

113533 

1646 

886467 

36 

25 

110873 

1616 

996351 

27 

114521 

1643 

885479 

35 

26 

-  111842 

1612 

996335 

27 

115507 

1639 

884493 

34 

27 

112809 

1608 

996318 

27 

116491 

1636 

883509 

33 

28 

113774 

'  1605 

996302 

28 

117472 

1632 

882528 

32 

29 

114737 

1601 

996285 

28 

118452 

1629 

881548  31 

30 

115698 

1597 

996269 

28 

119429 

1625 

880571 

30 

31 

9.116656 

1594 

9.996252 

28 

9.120404 

1622 

10.879596 

29 

32 

117613 

1590 

996235 

28 

121377 

1618 

878623 

"28 

33 

118567 

1587 

996219 

28 

122348 

1615 

877652 

27 

34 

119519 

1583 

996202 

28 

123317 

1611 

876683 

26 

35 

120469 

1580 

996185 

28 

124284 

1607 

875716 

25 

36 

121417 

1576 

996168 

28 

125249 

1604 

874751 

24 

37 

122362 

1573 

996151 

28 

126211 

1601 

873789 

23 

38 

123306 

1569 

996134 

28 

127172 

1597 

872828 

22 

39 

124248 

1566 

996117 

28 

128130 

1594 

871870 

21 

40 

125187 

1562 

996100 

>28 

129087 

1591 

870913 

20 

41 

9.126125 

1559 

9.996083 

29 

9.130041 

1587 

10.869959 

19 

42 

127060 

1556 

996066 

29 

130994 

1584 

869006 

18 

43 

127993 

1552 

996049 

29 

131944 

1581 

868056 

17 

4-1 

128925 

1549 

996032 

29 

132893 

1577 

867107 

16 

45 

129854 

1545 

996015 

29 

133839 

1574 

866161 

16 

46 

130781 

1542 

995998 

29 

134784 

1571 

865216 

14 

47 

131706 

1539 

995980 

29 

135726 

1567 

864274 

13 

48 

132630 

1535 

995963 

29 

136667 

1564 

863333 

12 

49 

133551 

1532 

995946 

29 

137605 

1561 

862395 

11 

50 

134470 

1529 

995928  29 

138542 

1558 

861458 

10 

51 

9.135387 

1525 

9.995911 

29 

9.139476 

1555 

10.860524 

9 

52 

136303 

1522 

995894 

29 

140409 

1551 

859591 

8 

53 

137216 

1519 

995876  j  29 

141340 

1548 

858660 

54 

138128 

1516 

995859  29 

142269 

1545 

857731 

( 

55 

139037 

1512 

995841  29 

143196 

1542 

856804 

{ 

56 

139944 

1509 

995823!  29 

144121 

1539 

855879 

t 

57 

140850 

1506 

995806 

29 

145044 

1535 

854956 

', 

58 

141754 

150S 

995788 

29 

145966 

1532 

854034 

' 

59 

142655 

1500 

995771 

29 

146885 

1529 

853115 

60 

143555 

1496 

995753 

29 

147803  1526 

852197 

1 

Cosine 

Sine         (.otcii.i'.         J   Itti.g    M. 

82  Degrees. 

D 

20 


(8  Degrees.;     A  TABLE  OF  LOGARITHMIC 


M. 

Sine      D.     Cosine   |  D.    Tang.   |   D.   |  Cotang.   | 

0 

9.143555 

1496 

9.995753 

30  9.147803 

1526 

I0.852197i  60 

I 

144453 

1493 

995735 

30  '   148718 

1523 

851282 

59 

2 

145349 

1490 

995717 

30 

149632 

1520 

850368 

58 

3 

146243 

1487 

*  995699 

30 

150544 

1517 

849456 

57 

4 

147136 

1484 

995681 

30 

151454 

1514 

848546 

56 

5 

148026 

1481 

995664 

30 

152363 

1511 

847637 

55 

c 

148915 

1478 

995646 

30 

153269 

1508 

846731 

54 

7 

149802 

1475 

995628 

30 

154174 

1505 

845826 

53 

8 

150686 

1472 

995610 

30 

155077 

1502 

844923 

52 

9 

151569 

1469 

995591 

30 

155978 

1499 

844022 

21 

10 

152451 

1466 

995573 

30 

156877 

1496 

843123 

50 

11 

9  153330 

1463 

9.995555 

30 

9.157775 

1493 

10.842225 

49 

12 

154208 

1460 

995537 

30 

158671 

1490 

841329 

48 

13 

155083 

1457 

995519 

30 

159565 

1487 

840435 

47 

14 

156957 

1454 

995501 

31 

160457 

1484 

839543 

46 

15 

156830 

1451 

995482 

31 

161347 

1481 

838653 

45 

16 

157700 

1448 

995464 

31 

162236 

1479 

837764 

44 

17 

158569 

1445 

995446 

31 

163123 

1476 

836877 

43 

18 

159435 

1442 

995427 

31 

164008 

1473 

835992 

42 

19 

160301 

1439 

995409 

31 

164892 

1470 

835108 

41 

20 

161164 

1436 

995390 

31 

165774 

1467 

834226 

40 

21 

9.162025 

1433 

9.995372 

31 

9.166654 

1464 

10.833346 

39 

22 

162885 

1430 

995353 

31 

167532 

1461 

832468 

38 

23 

163743 

1427 

995334 

31 

168409 

1458 

831591 

37 

24 

164600 

1424 

995316 

31 

169284 

1455 

830716 

36 

25 

165454 

1422 

995297 

31 

170157 

1453 

829843 

35 

26 

166307 

1419 

995278 

31 

171029 

1450 

828971 

34 

27 

167159 

1416 

995260 

31 

171899 

1447 

828101 

33 

28 

168008 

1413 

995241 

32 

172767 

1444 

827233 

32 

29 

168856 

1410 

995222 

32 

173634 

1442 

826366 

31 

30 

169702 

1407 

995203 

32 

174499 

1439 

825501 

30 

31 

9.170547 

1405 

9.995184 

32 

9.175362 

1436 

10.824638 

29 

32 

171389 

1402 

995165 

32 

176224 

1433 

823776 

28 

33 

172230 

1399 

995146 

32 

177084 

1431 

822916 

27 

34 

173070 

1396 

995127 

32 

177942 

1428 

822058 

26 

35 

173908 

1394 

995108 

32 

178799 

1425 

821201 

25 

36 

174744 

1391 

995089 

32 

179655 

1423 

820345 

24 

37 

175578 

1388 

995070 

32 

180508 

1420 

819492 

23 

38 

176411 

1386 

995051 

32 

181360 

1417 

818640 

22 

39 

177242 

1383 

995032 

32 

182211 

1415 

817789 

21 

40 

178072 

1380 

995013  32 

183059 

1412 

816941 

20 

41 

9.178900 

1377 

9.994993 

32 

9.183907 

1409 

10.816093 

19 

42 

179726 

1374 

994974 

32 

184752 

1407 

815248 

18 

43 

180551 

1372 

994955 

32 

185597 

1404 

814403 

17 

44 

181374 

1369 

994935 

32 

186439 

1402 

813561 

16 

45 

182196 

1366 

994916 

33 

187280 

1399 

812720 

15 

46 

183016 

1364 

994896 

33 

188120 

1396 

811880 

14 

47 

183834 

1361 

994877 

33 

188958 

1393 

811042 

13 

48 

184651 

1359 

994857 

33 

.  189794 

1391 

810206 

12 

49 

185466 

1356 

994838 

33 

190629 

1389 

809371 

11 

50 

186280 

1353 

994818 

33 

191462 

1386 

808538 

10 

51 

9.187092 

1351 

9.994798 

33 

9.192294 

1384 

10.807706 

9 

52 

187903 

1348 

994779 

33 

193124 

1381 

80G876 

8 

53 

188712 

1346 

994759 

33 

193953 

1379 

806047 

7 

54 

189519 

1343 

994739 

33 

194780 

1376 

805220 

6 

55 

190325 

1341 

994719 

33 

195606 

1374 

804394 

5 

56 

191130 

1338 

994700 

33 

196430 

1371 

803570 

4 

57 

191933 

1336 

994680 

33 

197253 

1369 

802747 

3 

58 

192734 

1333 

994660 

33 

198074 

1366 

801926 

2 

59 

193534 

1330 

994640 

33 

198894 

1364 

801106 

7 

60 

194332 

1328 

994620 

33 

199713 

1361 

800287 

0 

Cosine   J 

Sine 

Cotang.  1 

Tanir.   |  M. 

81  Degrees. 


SINES   AT^D    TANGF-NTS.       (9    Degrees.; 


27 


M.    Sine      D.     Cosine   |  D.    Tang.      D.     Comng. 

0 

9.194332 

1328 

9.994620 

33 

9.199713 

1361 

10.800287 

60 

195129 

1326 

994600 

33 

200529 

1359 

799471 

59 

2 

195925 

1323 

994580 

33 

201345 

1356 

798655 

5S 

3 

196719 

1321 

994660 

34 

202159 

1354 

797841 

57 

4 

197511 

1318 

994540 

34 

202971 

1352 

797029 

56 

5 

198302 

1316 

994519 

34 

203782 

1349 

796218 

55 

6 

199091 

1313 

994499 

34 

204592 

1347 

795408 

54 

7 

199879 

1311 

994479 

34 

205400 

1345 

794600 

53 

8 

200666 

1308 

994459 

34 

206207 

1342 

793793 

52 

9 

201451 

1306 

994438 

34 

207013 

1340 

792987 

51 

10 

202234 

1304 

994418 

34 

207817 

1338 

792,183 

50 

11 

9.203017 

1301 

9.994397 

34 

9.208619 

1335 

10.791381 

49 

12 

203797 

1299 

994377 

34 

209420 

1333 

790580 

48 

13 

204577 

1296 

994357 

34 

210220 

1331 

78978%|  47 

14 

205354 

1294 

994336 

34 

211018 

1328 

788982 

46 

15 

206131 

1292 

994316 

34 

211815 

1326 

788185 

45 

16 

206906 

1289 

994295 

34 

212611 

1324 

787389 

44 

17 

207679 

1287 

994274 

35 

213405 

1321 

786595 

43 

18 

208452 

1285 

994254 

35 

214198 

1319 

785802 

42 

19 

209222 

1282 

994233 

35 

214989 

1317 

785011 

41 

20 

209992 

1280 

994212 

35 

215780 

1315 

784220 

40 

21 

9.210760 

1278 

9.994191 

35 

9.216568 

1312 

10.783432 

39 

22 

211526 

1275 

994171 

35 

217356 

1310 

782644 

38 

23 

212291 

1273 

994150 

35 

218142 

1308 

781858 

37 

24 

213055 

1271 

994129 

35 

218926 

1305 

781074 

36 

25 

213818 

1268 

994108 

35 

219710 

1303 

780290 

35 

26 

214579 

1266 

994087 

35 

220492 

1301 

779508 

34 

27 

215338 

1264 

994066 

35 

221272 

1299 

778728 

33 

28 

216097 

1261 

994045 

35 

222052 

1297 

777948 

32 

29 

216854 

1259 

994024 

35 

222830 

1294 

777170 

31 

30 

217609 

1257 

994003 

35 

223606 

1292 

776394 

30 

31 

9.218363 

1255 

9.993981 

35 

9.224382 

1290 

10.775618 

29 

32 

219116 

1253 

993960 

35 

225156 

1288 

774844 

28 

33 

219868 

1250 

993939 

35 

225929 

1286 

774071 

27 

34 

220618 

1248 

993918 

35 

226700 

1284 

7*73300 

26 

35 

221367 

1246 

993896 

36 

227471 

1281 

772529 

25 

36 

222115 

1244 

993875 

36 

228239 

1279 

771761 

24 

37 

222861 

1242 

993854 

36 

229007 

1277 

770993!  23 

38 

223606 

1239 

993832 

36 

219773 

1275 

770227 

22 

39 

224349 

1237 

993811 

36 

230539 

1273 

769461 

21 

40 

225092 

1235 

993789 

36 

231302 

1271 

768698 

20 

41 

9.225833 

1233 

9.993768 

36 

9.232065 

1269 

10.767935 

19 

42 

226573 

1231 

993746 

36 

232826 

1267 

767174 

18 

43 

227311 

1228 

993725 

36 

233586 

1265 

766414 

17 

44 

228048 

122C 

993703 

36 

234345 

1262 

765655 

16 

45 

228784 

1224 

993681 

36 

235103 

1260 

764897 

15 

46 

229518 

1222 

993660 

36 

235859 

1258 

764141 

14 

47 

230252 

1220 

993638 

36 

236614 

1256 

763386 

13 

48 

230984 

1218 

993616 

36 

237368 

1254 

762632 

12 

49 

231714 

1216 

993594 

37 

238120 

1252 

761880 

11 

50 

232444 

1214 

993572 

37 

238872 

1250 

761128 

10 

51 

9.233172 

1212 

9.993550 

37 

9.239622 

1248 

10.760378 

9 

52 

233899 

1209 

993528 

37 

240371 

1246 

759629 

8 

53 

234625 

1207 

993506 

37 

241118 

1244 

758882 

7 

54 

235349 

1205 

993484 

37 

241865 

1242 

758135 

6 

55 

236073 

1203 

993462 

37 

242610 

1240 

757390 

5 

56 

236795 

1201 

993440 

37 

243354 

1238 

756646 

4 

57 

237515 

1199 

993418 

37 

244097 

1236 

755903 

3 

58 

238235 

1197 

993396 

37 

244839 

1234 

755161 

2 

59 

238953 

1195 

993374 

37 

245579 

1232 

754421 

1 

60 

2396701  1193 

993351 

37 

246319 

1230 

753681 

0 

|  Cosine   |       |   Sine   |     Cotang.            Tang.   JM. 

80  Degrees. 


28 


(10  Degrees.)     A  TABLE  OF  LOGABITHMIC 


M.    Sine    |   D.     Cosine 

D. 

Tang. 

D. 

Cotansr.  [ 

0 

9.239670 

1193 

9.993351 

37 

9.246319 

1230 

10.753681 

60 

1 

240386 

1191 

993329 

37 

247057 

1228 

752943 

59 

2 

241101 

1189 

993307 

37 

247794 

1226 

752206 

58 

3 

241814 

1187 

993285 

37 

248530 

1224 

751470 

57 

4 

242526 

1185 

993262 

37 

249264 

1222 

750736 

56 

5 

243237 

1183 

993240 

37 

249998 

1220 

750002 

55 

6 

243947 

1181 

993217 

38 

250730 

121S 

749270 

54 

7 

244656 

1179 

993195 

38 

251461 

1217 

748539 

53 

8 

245363 

1177 

993172 

38 

252191 

1215 

747809 

52 

9 

246069 

1175 

993149 

38 

252920 

1213 

747080 

51 

10 

246775 

1173 

993127 

38 

253648 

1211 

746352 

50 

11 

9  .  247478 

1171 

9.993104 

38 

9.254374 

1209 

10.745626 

49 

12 

248181 

1169 

993081 

38 

255100 

1207 

744900 

48 

13 

*  248883 

1167 

993059 

38 

255824 

1205 

744176 

47 

14 

249583 

1165 

993036 

38 

256547 

1203 

743453 

46 

15 

250282 

1163 

993013 

38 

257269 

1201 

742731 

45 

16 

250980 

1161 

992990 

38 

25T990 

1200 

742010 

44 

17 

251677 

1159 

992967 

38 

258710 

1198 

741290 

43 

18 

252373 

1158 

992944 

38 

259429 

1196 

740571 

42 

19 

253067 

1156 

992921 

38 

260146 

1194 

739854 

41 

20 

253761 

1154 

992898 

38 

260863 

1192 

739137 

40 

21 

9  .  254453 

1152 

9.992875 

38 

9.261578 

1190 

10.738422 

39 

22 

255144 

1150 

992852 

38 

262292 

1189 

737708 

38 

23 

255834 

1148 

992829 

39 

263005 

1187 

736995 

37 

24 

256523 

1146 

992806 

39 

263717 

1185 

736283 

36 

25 

257211 

1144 

992783 

39 

264428 

1183 

735572 

35 

26 

257898 

1142 

992759 

39 

265138 

1181 

734862 

34 

27 

258583 

1141 

992736 

39 

265847 

1179 

734153 

33 

28 

259268 

1139 

992713 

39 

266555 

1178 

733445 

32 

29 

259951 

1137 

992690 

39 

267261 

1176 

732739 

31 

30 

260633 

1135 

•  992666 

39 

267967 

1174 

732033 

30 

31 

9.261314 

1133 

9.992643 

39 

9.268671 

1172 

10.731329 

29 

32 

261994 

1131 

992619 

39 

269375 

1170 

730625 

28 

33 

262673 

1130 

992596 

39 

270077 

1169 

729923 

27 

34 

263351 

1128 

992572 

39 

270779 

1167 

729221 

26 

35 

264027 

1126 

992549 

39 

271479 

1165 

728521 

25 

36 

264703 

1124 

992525 

39 

272178 

1164 

727822 

24 

37 

265377 

1122 

992501 

39 

272876 

1162 

727124 

23 

38 

266051 

1120 

992478 

40 

273573 

1160 

726427 

22 

39 

266723 

1119 

992454 

40 

274269 

1158 

725731 

21 

40 

267395 

1117 

992430 

40 

274964 

1157 

725036 

20 

41 

9.268065 

1115 

9.992406 

40 

9.275658 

1155 

10.724342 

19 

42 

268734 

1113 

992382 

40 

276351 

1153 

723649 

18 

43 

269402 

1111 

992359 

40 

277043 

1151 

722957 

17 

44 

270069 

1110 

992335 

40 

277734 

1150 

722266 

16 

45 

270735 

1108 

992311 

40 

278424 

1148 

721576 

15 

46 

271400 

1106 

992287 

40 

279113 

1147 

720887 

14 

47 

272064 

1105 

992263 

40 

279801 

1145 

720199 

13 

48 

272726 

1103 

992239 

40 

280488 

1143 

719512 

12 

49 

273388 

1101 

992214 

40 

281174 

1141 

718826 

11 

50 

274049 

1099 

992190 

40 

281858 

1140 

718142 

10 

51 

9.274708 

1098 

9.992166 

40 

9.282542 

1138 

10.717458 

9 

52 

275367 

1096 

992142 

40 

283225 

1136 

716775 

8 

53 

276024 

1094 

992117 

41 

283907 

1135 

716093 

7 

54 

276681 

1092 

992093 

41 

284588 

1133 

715412 

6 

55 

277337 

1091 

992069 

41 

285268 

1131 

714732 

5 

56 

277991 

1089 

992044 

41 

285947 

1130 

714053 

4 

57 

278644 

1087 

992020 

41 

286624 

1128 

713376 

3 

58 

279297 

1086 

991996 

41 

287301 

1126 

712699 

2 

59 

279948 

1084 

991971 

41 

287977 

1125 

712023 

1 

60 

280599 

1082 

991947 

41 

288652 

1123 

711348 

0 

Cosine 

Sine 

)  Cotang.  | 

Tang. 

M. 

79  Degrees. 


SINES  AND  TANGENTS.     ( 1 1   Degrees.) 


29 


>l.    Sine 

P.   |  Citsii*    l>.  |   Tana.   |   D.     Cota-ia.   | 

0 

9.  280599 

1082 

9.991947 

41 

9.288652 

1123 

10.711348  60 

1 

281248 

1081 

991922 

41 

289326 

1122 

710674 

59 

2 

281897 

1079 

991897 

41 

289999 

1120 

710001 

58 

3 

282544 

1077 

991873 

41 

290671 

1118 

709329 

57 

4 

283190 

1076 

991848 

41 

291342 

1117 

708658 

56 

5 

283836 

1074 

991823 

41 

292013 

1115 

707987 

55 

6 

284480 

1072 

991799 

41 

292682 

1114 

707318 

54 

7 

285124 

1071 

991774 

42 

293350 

1112 

706650 

53 

8 

285766 

1069 

991749 

42 

294017 

1111 

705983 

52 

9 

286408 

1067 

991724 

42 

294684 

1109 

705316 

51 

10 

287048 

1066 

991699 

42 

295349 

1107 

704651 

50 

11 

9.287687 

1064 

9.991674 

42 

9.296013 

1106 

10.703987 

49 

12 

288326 

1063 

991649 

42 

296677 

1104 

703323 

48 

13 

238964 

1061 

991624 

42 

297339 

1103 

70266  1 

47 

14 

289600 

1059 

991599 

42 

298001 

1101 

701999 

46 

15 

290236 

1058 

991574 

42 

2J8662 

1100 

701338 

45 

16 

290870 

1056 

991549 

42 

299322 

1098 

700678 

44 

17 

291504 

1054 

991524 

42 

299980 

1096 

700020 

43 

13 

292137 

1053 

991498 

42 

300638 

1095 

699362 

42 

19 

292768 

1051 

991473 

42 

301295 

1093 

698705 

41 

20 

293399 

1050 

991448 

42 

301951 

1092 

698049 

40 

21 

9  .  294029 

1043 

9.991422 

42 

9.302607 

1090 

10.697393 

39 

22 

294658 

1046 

991397 

42 

303261 

1089 

696739 

38 

23 

295286 

1045 

991372 

43 

303914 

1087 

696086 

37 

24 

295913 

1043 

991346 

43 

304567 

1086 

695433 

36 

25 

296539 

1042 

991321 

43 

305218 

1084 

694782 

35 

26 

297164 

1040 

991295 

43 

305869 

1083 

694131 

34 

27 

297788 

1039 

991270 

43 

306519 

1081 

693481 

33 

23 

298412 

1037 

991244 

43 

307168 

1080 

692832 

32 

23 

299034 

1036 

991218 

43 

307815 

1078 

692185 

31 

30 

299655 

1034 

991193 

43 

308463 

1077 

691537 

30 

31 

9.300276 

1032 

9.991167 

43 

9.309109 

1075 

10.690891 

29 

32 

300895 

1031 

991141 

43 

309754 

1074 

690246 

28 

33 

301514 

1029 

991115 

43 

310398 

1073 

689602 

27 

34 

302132 

1028 

99'1090 

43 

811042 

1071 

6889G8 

26 

35 

302748 

1026 

991064 

43 

311685 

1070 

688315 

25 

36 

303364 

1025 

991038 

43 

312327 

1068 

687673 

24 

37 

303979 

1023 

991012 

43 

312967 

1067 

687033 

23 

33 

304593 

1022 

990986 

43 

313608 

1065 

686392 

22 

39 

305207 

1020 

990960 

43 

314247 

1064 

685753 

21 

40 

305819 

1019 

990934 

44 

314885 

1062 

685115 

20 

41 

9.306430 

1017 

9.990908 

44 

9.315523 

1061 

10.684477 

19 

42 

307041 

1016 

990882 

44 

316159 

1060 

683841 

18 

43 

307650 

1014 

990855 

44 

316795 

1058 

683205 

17 

44 

308259 

1013 

990829 

44 

317430 

1057 

682570 

16 

45 

308867 

1011 

990803  44 

318064 

1055 

681936 

15. 

46 

309474 

1010 

990777  44 

318697 

1054 

681303 

14 

47 

310080 

1008 

990750'  44 

319329 

1053 

680671 

13 

48 

310685 

1007 

990724;  44 

319961 

1051 

680039 

12 

49 

311289 

1005 

990697  44 

320592 

1050 

679408 

11 

50 

311893 

1004 

990671  44 

321222 

1048 

678778 

10 

51 

9.312495 

1003  19.990644  44 

9.  321851  i  1047 

10.678149 

9 

52 

313097 

1001 

990618  44 

322479*  1045 

677521 

8 

53 

313698 

1000 

990591  44 

323106  1044 

676894 

7 

54 

314297 

998 

990565  44 

323733  1043 

676267 

6 

55 

314897 

997 

990538  44 

324358   1041 

675642 

5 

56 

315495 

996 

990511  45 

324983  1040 

675017 

4 

57 

316092 

994 

9904.S5  45 

325607  1039 

674393 

3 

58 

316689   993 

990458  45    326231   1037 

673769  2 

59 

317284   .191  I   990431  45   326853  1036 

673147  1 

GO 

317879   990  !  ,990404  45   327475  1035 

672525  0 

j   Cosine 

1   Sine       |  Cotartg. 

|   Tang.   ( 

78  Degrees. 


30 


(12  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.     Sine   |   D.   |   Conine   |  D.  |   Tang. 

D. 

Cotanji.  | 

0 

9.317879 

990 

9.990404 

45 

9.327474 

1035 

10.672526 

60 

1 

318473 

988 

990378 

45 

328095 

1033 

671905 

59 

2 

319066 

987 

990351 

45 

328715 

1032 

671285 

58 

3 

319658 

986 

990324 

45 

329334 

1030 

6706.66 

57 

4 

320249 

984 

990297 

45 

329953 

1029 

670047 

56 

5 

320840 

983 

990270 

45 

330570 

1028 

669430 

55 

6 

321430 

982 

990243 

45 

331187 

1026 

668813 

54 

7 

322019 

980 

990215 

45 

331803 

1025 

668197 

53 

8 

322607 

979 

990188 

45 

332418 

1024 

667582 

52 

9 

323194 

977 

990161 

45 

333033 

1023 

666967 

51 

10 

323780 

976 

990134 

45 

333646 

1021 

666354 

50 

11 

9.324366 

975 

9.99~0107 

46 

9.334259 

1020 

10.665741 

49 

12 

324950 

973 

990079 

46 

334871 

1019 

665129 

48 

13 

325534 

972 

990052 

46 

335482 

1017 

664518 

47 

14 

326117 

970 

990025 

46 

336093 

1016 

663907 

46 

15 

326700 

969 

981,997 

46 

336702 

1015 

663298 

45 

16 

327281 

968 

9899-70 

46 

337311 

1013 

662689 

44 

17 

327862 

966 

989942 

46 

•337919 

1012 

662081 

43 

18 

328442 

965 

989915 

46 

338527 

1011 

661473 

42 

19 

329021 

964 

989837 

46 

339133 

1010 

660867 

41 

20 

329599 

962 

989860 

46 

339739 

1008 

660261 

40 

21 

9.330176 

961 

9.989832 

46 

9.340344 

1007 

10.659656 

39 

22 

330753 

960 

989804 

46 

340948 

1006 

659052 

38 

23 

331329 

958 

989777 

46 

341552 

1004 

658448 

37 

24 

331903 

957 

989749 

47 

342155 

1003 

657845 

36 

25 

332478 

956 

989721 

47 

342757 

1002 

657243 

35 

26 

333051 

954 

989693 

47 

343358 

1000 

656642 

34 

27 

333624 

953 

989665 

47 

843958 

999 

656042 

33 

28 

334195 

952 

989637 

47 

344558 

998 

655442 

32 

29 

334766 

950 

989609 

47 

345157 

997 

654843 

31 

30 

335337 

949 

989582 

47 

345755 

996 

654245 

30 

31 

9.335906 

948 

9  .  989553 

47 

9.346353 

994 

10.653647 

29 

32 

336475 

946 

989525 

47 

346949 

993 

653051 

28 

33 

337043 

945 

989497 

47 

347.545 

992 

652455 

27 

34 

337610 

944 

989469 

47 

348141 

991 

651859 

26 

35 

338176 

943 

989441 

47 

348735 

990 

651265 

25 

36 

338742 

941 

989413 

47 

349329 

988 

650671 

24 

37 

339306 

940 

989384 

47 

349922 

987 

650078 

23' 

38 

339871 

939 

989356 

47 

350514 

986 

64948C 

22 

39 

340434 

937 

989328 

47 

351106 

985 

64P894 

21 

40 

340996 

936 

989300 

47 

351697 

983 

648303 

20 

41 

9.341558 

935 

9.989271 

47 

9.352287 

982 

10.647713 

19 

42 

342119 

934 

989243 

47 

352876 

981 

647124 

18 

43 

342679 

932 

989214 

47 

353465 

980 

646535 

17 

44 

343239 

931 

989186 

47 

354053 

979 

645947 

16 

45 

343797 

930 

989157 

47 

354640 

977 

645360 

15 

46 

344355 

929 

989128 

48 

355227 

976 

644773 

14 

47 

344912 

927 

989100 

48 

355813 

975 

644187 

13 

48 

345469 

926 

989071 

48 

356398 

974 

643602 

12 

49 

346024 

925 

989042 

48 

356982 

973 

643018 

11 

.50 

346579 

924 

989014 

48 

357566 

971 

642434 

10 

51 

9.347134 

922 

9.988985 

48 

9.358149 

970 

10.641851 

9 

52 

347687 

921 

988956 

48 

358731 

969 

641269 

8 

53 

348240 

920 

988927 

48 

359313 

968 

640687 

7 

54 

348792 

919 

988898 

48 

359893 

967 

640107 

6 

55 

349343 

917 

988869 

48 

360474 

966 

639526 

5 

56 

349893 

916 

988840 

48 

361053 

965 

638947 

4 

57 

350443 

915 

988811 

49 

361632 

963 

638368 

3 

58 

350992 

914 

988782 

49 

362210 

962 

637790 

2 

59 

351540 

913 

988753 

49 

362787 

961 

637213 

] 

60 

352088 

911 

988724 

49 

363364 

960 

63663fi 

0 

Cosine   1       1   Sine   I    I  Colang. 

II                  II 

Tang   |  M. 

77  Degrees. 


SINES  AND  TANGENTS.     (13  Degrees.) 


31 


M.  |   Sine      D.     Cosine    I).    Ta-iti.      1>. 

Oiianii. 

0  |  9.352088 

911 

9  .  988724 

49 

9.3H33H4 

960 

10.636630 

60 

1 

352635 

910 

988695 

49 

363940 

959 

636060 

59 

2 

353181 

909 

983666 

49 

364515 

958 

635485 

58 

3 

353726 

908 

988636 

49 

365090 

957 

634910 

57 

4 

354271 

907 

988607 

49 

365664 

955 

6343361  56 

5 

354815 

905 

988578 

49 

366237 

954 

633763 

55 

6 

355353 

904 

988548 

49 

366810 

953 

633190 

54 

7 

355901 

903 

988519 

49 

367382 

952 

632618 

53 

8 

356443 

902 

988489 

49 

367953 

951 

632047 

52 

9 

356934 

901 

988460 

49 

368524 

950 

631476 

51 

10 

357524 

899 

983430 

49 

369094 

949 

630906 

50 

11 

9.358064 

898 

9.988401 

49 

9.369663 

948 

10.630337 

49 

12 

358603 

897 

988371 

49 

370232 

946 

629768 

48 

13 

359141 

896 

988342 

49 

370799 

945 

629201 

47 

14 

359673 

895 

988312 

50 

371367 

944 

628633 

46 

15 

360215 

893 

988282 

50 

371933 

943 

623067 

45 

16 

360752 

892 

988252 

50 

372499 

942 

627501 

44 

17 

361287 

891 

988223 

50 

373064 

941 

626936 

43 

18 

361822 

890 

988193 

50 

373629 

940 

626371 

42 

19 

362356 

889 

988163 

50 

374193 

939 

625807 

41 

20 

362889 

888 

988133 

50 

374756 

938 

625244 

40 

21 

9.363422 

887 

9.988103 

50 

9.375319 

937 

10762468  1 

39 

22 

363954 

885 

988073 

50 

375881 

935 

624119 

38 

23 

3T>4485 

884 

988043 

50 

376442 

934 

623558 

37 

24 

365016 

883 

988013 

50 

377003 

933 

622997 

36 

25 

365546 

8S2 

987983 

50 

377563 

932 

622437 

35 

26. 

366075 

881 

987953 

50 

378122 

931 

621878 

34 

27 

366604 

880 

987922 

50 

378681 

930 

621319 

33 

23 

367131 

879 

987892 

50 

379239 

929 

620761 

32 

29 

367659 

877 

987862 

50 

379797 

928 

620203 

31 

30 

368185 

876 

987832 

51 

380354 

927 

619646 

30 

31 

9.363711 

875 

9.987801 

51 

9.380910 

926 

10.619090 

29 

32 

369236 

874 

987771 

51 

381466 

925 

618534 

28 

33 

369761 

873 

987740 

51 

382020 

924 

617980 

27 

34 

370285 

872 

9*87710 

51 

382575 

923 

617425 

26 

35 

370808 

871 

987679 

51 

383129 

922 

616871 

25 

36 

371330 

870 

987649 

51 

383682 

921 

616318 

24 

37 

371852 

869 

987618 

51 

384234 

920 

615766 

23 

38 

372373 

867 

987588 

51 

384786 

919 

615214 

22 

39 

372894 

866 

987557 

51 

385337 

918 

614663 

581 

40 

373414 

865 

987526 

51 

385888 

917 

614112 

20 

41 

9.373933 

864 

9.987496 

51 

9.386433 

915 

10.613562 

19 

42 

374452 

863 

987465 

51 

386987 

914 

613013 

18 

43 

374970 

862 

987434 

51 

387536 

913 

612464 

17 

44 

375487 

861 

987403 

52 

838084 

912 

611916 

16 

45 

376003 

860 

987372 

52 

388631 

911 

611369 

15 

46 

376519 

859 

987341 

52 

389178 

910 

610822 

14 

47 

377035 

858 

997310 

52 

389724 

909 

610276 

13 

43 

377549 

857 

987279 

52 

390270 

908 

609730 

12 

49 

3780G3 

856 

987248 

52 

390815 

907 

609185 

11 

50 

378577 

854 

987217 

52 

391360 

906 

603640 

10 

51 

9.379089 

853 

9.987186 

52 

9.391903 

905 

10.608097 

9 

52 

379601 

852 

987155 

52 

392447 

904 

607553 

8 

53 

380113 

851 

937124 

52 

392989 

903 

607011 

7 

54 

380624 

850 

987092 

52 

393531 

902 

606469 

6 

55 

331134 

849 

987061 

52 

394073 

901 

605927 

5 

56 

381643 

848 

987030 

52 

394614 

900 

605386 

4 

57 

382152 

847 

986998 

52 

395154 

899 

604846 

3 

58 

382661 

846 

986967 

52 

395694 

898 

604306 

2 

59 

383168 

845 

986936 

52 

396233 

897 

603767 

1 

60 

383675 

844 

986904 

52 

396771 

896 

603229 

0 

j  C..si»«            Sine 

Cot  IMC. 

lanp. 

M. 

76  Degrees. 


32  (14  Degrees.)     A  TABLE  or  LOGARITHMIC 


M. 

Sine 

I).   |  Cosine 

I).  |   Tang.     D. 

C'otang. 

0 

9.383675 

844 

9.986904 

52 

9.396771 

896 

10.603229 

60 

1 

384182 

843 

986873 

53 

397309 

896 

602691 

59 

2 

384687 

842 

986841 

53 

397846 

895 

602154 

58 

3 

385192 

841 

986809 

53 

398383 

894 

601617 

57 

4 

385697 

840 

986778 

53 

398919 

893 

601081 

56 

5 

386201 

839 

986746 

53 

399455 

892 

600545 

55 

6 

386704 

838 

986714 

53 

399990 

891 

600010 

54 

7 

387207 

837 

986683 

53 

400524 

890 

599476 

53 

8 

387709 

836 

986651 

53 

401058 

889 

598942 

52 

9 

388210 

835 

986619 

53 

401591 

888 

598409 

51 

10 

388711 

834 

986587 

53 

402124 

887 

597876 

50 

11 

9.389211 

833 

9.986555 

53 

9.402656 

886 

10.597344 

49 

12 

389711 

832 

986523 

53 

403187 

885 

596813 

48 

13 

390210 

831 

986491 

53 

403718 

884 

596282 

47 

14 

390708 

830 

986459 

53 

404249 

883 

595751 

46 

15 

391206 

828 

986427 

53 

404778 

882 

595222 

45 

16 

391703 

827 

986396 

53 

405308 

881 

594692 

44 

17 

392199 

826 

986363 

54 

405836 

880 

594164 

43 

18 

392695 

825 

986331 

54 

406364 

8,79 

593636 

42 

19 

393191 

824 

986299 

54 

406892 

878 

593108 

41 

20 

393685 

823 

986266 

54 

407419 

877 

592581 

40 

21 

9.394179!  822 

9.986234 

54 

9.407945 

876 

10.592055 

39 

22 

394673   821 

986202 

54 

408471 

875 

591529 

38 

23 

395166 

820 

986169 

54 

408997 

874 

591003 

37 

24 

395658 

819 

986137 

54 

409521 

874 

590479 

36 

S5 

396150 

816 

986104 

54 

410045 

873 

589955 

35 

26 

396641 

817 

986072 

54 

410569 

872 

589431 

34 

27 

397132 

17 

986039 

54 

411092 

871 

588908 

33 

28 

397621 

816 

986007 

54 

411615 

870 

588385 

32 

29 

398111 

815 

985974 

54 

412137 

869 

587863 

31 

30 

398600 

814 

985942 

54 

412658 

868 

587342 

30 

31 

9.399088 

813 

9.985909 

55 

9.413179 

867 

10.586821 

29 

32 

399575 

812 

985876 

p  c 

413699 

866 

586301 

28 

33 

400062 

811 

985843 

55 

414219 

865 

585781 

27 

34 

400549 

810 

985811 

55 

414738 

864 

585262 

26 

35 

401035 

809 

9S5778 

55 

415257 

864 

584743 

25 

36 

401520 

808 

985745 

55 

415775 

863 

584225 

24 

37 

402005 

807 

985712 

55 

416293 

862 

583707 

23 

38 

402489 

806 

985679 

55 

416810 

861 

583190 

22 

39 

402972 

805 

985646 

55 

417326 

860 

582674 

21 

40 

403455 

804 

985613 

55 

417842 

859 

582158 

20 

41 

9.403938 

803 

9.985580 

55 

9.418358 

858 

10.581642 

19 

42 

404420 

802 

985547 

55 

418873 

857 

581127 

18 

43 

404901 

801 

985514 

55 

419387 

856 

580613 

17 

44 

405382 

800 

985480 

55 

419901 

855 

580099 

16 

45 

405862 

799 

985447 

55 

420415 

855 

579585 

15 

46 

406341 

798 

985414 

56 

420927 

854 

579073 

14 

47 

406820 

797 

985380 

56 

421440 

853 

578560 

13 

48 

407299 

796 

985347 

56 

421952 

852 

578048 

12 

49 

407777 

795 

985314 

56 

422463 

851 

57753? 

11 

50 

408254 

794 

985280 

56 

422974 

850 

577026 

10 

51 

9.408731 

794 

9.985247 

56 

9.423484 

849 

10.576510 

9 

52 

409207 

793 

985213 

56 

423993 

848 

576007 

8 

53 

409682 

792 

985180 

56 

424503 

848 

575497 

7 

54 

4101571  791 

985146 

56 

425011 

847 

574989 

6 

55 

410632!  790 

985113 

56 

425519 

846 

574481 

5 

56 

4111061  789 

985079 

56 

426027 

845 

573973 

4 

57 

411579!  788 

985045 

56 

426534 

844 

573466 

3 

58 

412052   787 

985011 

56 

427041 

843 

572959 

2 

59 

412524   786 

984978 

56 

427547 

843 

572453 

1 

60 

412996'  785 

9S4944 

56 

428052 

842 

571  94« 

0 

Cosine 

Sine 

|  Coung.  | 

Tang    [  M. 

75  Degrees. 


SINKS    AND    TANGENTS.         (15DegreeS.) 


33 


M.    Sine      D.   |   Cosine   |  D.    Tarn.'-  |   D.     Cotaiie.  | 

0 

9.412996 

785 

9.984944 

57 

<K  428053 

842 

10.571948 

60 

1 

413467 

784 

984910 

57 

428557 

841 

571443 

59 

2 

413938 

783 

984876 

57 

429062 

840 

570938 

58 

3 

414408 

783 

984842 

57 

429566 

839 

570434 

57 

4 

414878 

782 

984808 

57 

430070 

838 

569930 

56 

5 

415347 

781 

984774 

57 

430573 

838 

1   569427 

55 

6 

4f5815 

780 

984740 

57 

431075 

837 

568925 

54 

7 

416283 

779 

934706 

57 

431577 

836 

568423 

53 

8 

416751 

778 

984672 

57 

432079 

835 

567921 

52 

9 

417217 

777 

984637 

57 

432580 

834 

567420 

51 

10 

417684 

770 

984603 

57 

4330SO 

833 

566920 

50 

11 

9.4F8150 

775 

9.984569 

57 

9.433580 

832 

10.566420 

49 

12 

418615 

774 

984535 

57 

434080 

832 

565920 

48 

13 

419079 

773 

984500 

57 

434579 

831 

565421 

47 

14 

419544 

773 

984466 

57 

435078 

830 

564922 

46 

15 

420007 

772 

984432 

58 

435576 

829 

564424 

45 

16 

420470 

771 

984397 

58 

436073 

828 

563927 

44 

17 

420933 

770 

984363 

58 

436570 

828 

563430 

43 

18 

421395 

769 

984328 

58 

437067 

827 

562933 

42 

19 

421857 

768 

984294 

58 

437563 

826 

562437 

41 

20 

422318 

767 

984259 

58 

438059 

825 

561941 

40 

21 

9  422778 

767 

9.984224 

58 

9.438554 

824 

10.561446 

39 

22 

423238 

766 

984190 

58 

439048 

823 

560952 

38 

23 

423697 

765 

984155 

58 

439543 

823 

560457 

37 

24 

424156 

764 

984120 

58 

440036 

822 

559964 

36 

25 

424615 

763 

984085 

58 

440529 

821 

559471 

35 

26 

425073 

762 

984050 

58 

441022 

820 

558978 

34 

27 

425530 

761 

984015 

58 

441514 

819 

558486 

33 

29 

425987 

760 

983981 

58 

442006 

819 

557994 

32 

29 

426-143 

760 

983946 

58 

442497 

818 

557503 

31 

30 

426899 

759 

983911 

58 

442988 

817 

557012 

30 

31 

9.427354 

758 

9.983875 

58 

9.443479 

816 

10.556521 

29 

32 

427809 

757 

983840 

59 

443968 

816 

556032 

28 

33 

428263 

756 

9S3805 

59 

444458 

815 

555542 

27 

34 

428717 

755 

983770 

59 

444947 

814 

555053 

26 

35 

429170 

754 

983735 

59 

445435 

813 

554565 

25 

36 

429623 

753 

983700 

59 

445923 

812 

554077 

24 

37 

430075 

752 

983664 

59 

446411 

812 

553589 

23 

38 

430527 

752 

983629 

59 

446898 

811 

553102 

22 

39 

430978 

751 

983594 

59 

447384 

810 

5526  tf 

21 

40 

431429 

750 

983558 

59 

447870 

809 

552130 

20 

41 

9.431879 

749 

9.983523 

59 

9.448356 

809 

107551644 

19 

42 

432329 

749 

983487 

59 

448841 

808 

551159 

18 

43 

432778 

748 

983452 

59 

449326 

807 

550674 

17 

44 

433226 

747 

9834161  59 

449810 

806 

550190 

16 

45 

433675 

746 

983381 

59 

450294 

806 

549706 

15 

46 

434122 

745 

983345 

59 

450777 

805 

549223 

14 

47 

434569 

744 

983309 

59 

451260 

804 

548740 

13 

48 

435016 

744 

983273 

60 

451743 

803 

•548257 

12 

49 

435462 

743 

983238 

60 

452225 

802 

547775 

11 

50 

435908 

742 

983202 

60 

452706 

802 

547294 

10 

51 

9.436353 

741 

9.983166 

60 

9.453187 

801 

10.546813 

9 

52 

436798 

740 

983130 

60 

453668 

800 

546332 

8 

53 

437242 

740 

983094 

60 

454*148 

799 

545852 

7 

54 

437686 

739 

983058 

60 

454628 

799 

545372 

6 

55 

438129 

738 

983022 

60 

455107 

798 

544893 

5 

56 

438572 

737 

982986 

60 

455586 

797 

514414 

4 

57 

439014 

736 

9829501  60 

456064 

796 

543936 

3 

58 

439456 

736 

982914  60 

456542 

796 

543458 

2 

59 

439897 

735 

982878 

60 

457019 

795 

542981 

1 

60 

440338 

734 

982842 

60 

457496 

794 

542504 

0 

Cosine         j   Sine 

Cotang.  |          Tang. 

M. 

74  Degrees. 

E 

34 


(16  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M. 

Sitio     D. 

Cosine  |  D.    Tang.     D.      Cotang.  | 

0 

9.440338 

734 

9.982842 

60 

9.457496 

794 

10.542504 

60 

1 

440778 

733 

982805 

60 

457973 

793 

542027 

59 

2 

441218 

732 

982769 

61 

458449 

793 

541551 

58 

3 

441658 

731 

982733 

61 

458925 

792 

541075 

57 

4 

442096 

731 

982696 

61 

459400 

791 

540600 

56 

5 

442535 

730 

982660 

61 

459875 

790 

54Q  125 

55 

6 

442973 

729 

982624 

61 

460349 

790 

539651 

54 

7 

443410 

728 

982587 

61 

460823 

789 

539177 

53 

8 

443847 

727 

982551 

61 

461297 

788 

538703 

52 

9 

444284 

727 

982514 

61 

461770 

788 

538230 

£1 

10 

444720 

726 

982477 

61 

462242 

787 

537758 

50 

11 

9.445155 

725 

9.982441 

61 

9.462714 

786 

10.537286 

49 

12 

445590 

724 

982404 

61 

463186 

785 

536814 

48 

13 

446025 

723 

982367 

61 

463658 

785 

536342 

47 

14 

446459 

723 

982331 

61 

464129 

784 

535871 

46 

15 

446S93 

722 

982294 

61 

464599 

783 

535401 

45 

16 

447326 

721 

982257 

61 

465069 

783 

634931 

44 

17 

447759 

720 

982220 

62 

465539 

782 

534461 

43 

18 

448191 

720 

982183 

62 

466008 

781 

533992 

42 

19 

448623 

719 

982146 

62 

466476 

780 

533524 

41 

20 

449054 

718 

982109 

62 

466945 

780 

533055 

40 

21 

9.449485 

717 

9.982072 

62 

9.467413 

779 

10.532587 

39 

22 

449915 

716 

982035 

62 

467880 

778 

532120 

38 

23 

450345 

716 

981998 

62 

468347 

778 

531*653 

37 

24 

450775 

715 

981961 

62 

468814 

777 

531186 

36 

25 

451204 

714 

981924 

62 

469280 

776 

530720 

35 

26 

451632 

713 

981886 

62 

469746 

775 

530254 

34 

27 

452060 

713 

981849 

62 

470211 

775 

529789 

33 

28 

452488 

712 

981812 

62 

470676 

774 

529324 

32 

29 

452915 

711 

981774 

62 

471141 

773 

528859 

31 

30 

453342 

710 

981737 

62 

471605 

773 

528395 

30 

31 

9.453768 

710 

9.981699 

63 

9.472068 

772 

10.527932 

29 

32 

454194 

709 

981662 

63 

472532 

771 

527468 

28 

33 

454619 

708 

981625 

63 

472995 

771 

527005 

27 

34 

455044 

707 

981587 

63 

473457 

770 

526543 

26 

35 

455469 

707 

981549 

63 

473919 

769 

526081 

25 

36 

455893 

70S 

981512 

63 

474381 

76*) 

525619 

24 

37 

456316 

705 

981474 

63 

474842 

768 

525158 

23 

38 

456739 

704 

981436 

63 

475303 

767 

524697 

22 

39 

457162 

704 

981399 

63 

475763 

767 

524237 

21 

40 

457584 

703 

981361 

63 

476223 

766 

523777 

20 

41 

9.458006 

702 

9.981323 

63 

9.476683 

765 

10.523317 

19 

42 

458427 

701 

981285 

63 

477142 

765 

522858 

18 

43 

458848 

701 

981247 

63 

477601 

764 

522399 

17 

44 

459268 

700 

981209 

63 

478059 

763 

521941 

16 

45 

459688 

699 

981171 

63 

478517 

763 

521483 

15 

46 

460108 

698 

981133 

64 

478975 

762 

521025 

14 

47 

460527 

698 

981095 

64 

479432 

761 

520568 

13 

48 

460946 

697 

981057 

64 

479889 

761 

520111 

12 

49 

461364 

696 

981019 

64 

480345 

760 

519655 

11 

50 

461782 

695 

980981 

64 

480801 

759 

519199 

10 

51 

9.462199 

695 

9.980942 

64 

9.481-257 

759 

10.518743 

9 

52 

462616 

694 

980904 

64 

481712 

758 

518288 

8 

53 

463032 

693  !   980866 

64 

482167 

757 

517833 

7 

54 

463448 

693 

980827 

64 

482621 

757 

517379 

6 

55 

46386' 

692 

980789 

64 

483075 

756 

516925 

5 

56 

464279 

691 

980750 

64 

483529 

755 

516471 

4 

57 

464694 

690 

980712 

64 

483982 

755 

516018 

3 

58 

465108 

690 

980673 

64 

484435 

754 

515565 

2 

59 

465522 

689 

980635  64 

484887 

753 

515113 

1 

60 

4659351  688 

98059b:  64 

485339 

753 

514fifil 

0 

Cosine            Sine 

|  Cotang. 

•»   Tang. 

M. 

73  Degrees. 


SINES  AND  TANGENTS.     (17  Degrees.) 


M.    Sine   |   D.     Cosine   |  D.  |   Tang.   |   D.     Cotang. 

0 

9.460935 

,688 

9.9SOf>96 

64 

9.485339 

755 

10.514661 

60 

1 

466348 

688 

980558 

64 

485791 

"£52 

514209 

C9 

2 

466761 

687 

980519 

65 

486242 

751 

513758 

58 

3 

467173 

686 

980480 

65 

486693 

751 

513307 

57 

4 

467585 

685 

980442 

65 

487143 

750 

512857 

56 

5 

467996 

685 

980403 

65 

487593 

749 

512407 

55 

6 

468407 

684 

980364 

65 

488043 

749 

511957 

54 

7 

468817 

683 

980325 

65 

488492 

748 

511508 

53 

8 

469227 

683 

980286 

65 

488941 

747 

511059 

52 

9 

469637 

682 

980247 

65 

489390 

747 

510610 

51 

10 

470046 

681 

980208 

65 

489838 

746 

510162 

50 

11 

9.470455 

680 

9.980169 

65 

9.490286 

746 

10.509714 

49 

12 

470863 

680 

980130 

65 

490733 

745 

509267 

48 

13 

471271 

679 

980091 

65 

491180 

744 

508820 

47 

14 

471679 

678 

980052 

65 

491627 

744 

508373 

46 

15 

472086 

678 

980012 

65 

492073 

743 

507927 

45 

16 

472492 

677 

979973 

65 

492519 

743 

507481 

44 

17 

472898 

676 

979934 

66 

492965 

742 

507035 

43 

18 

473304 

676 

979895 

66 

493410 

741 

506590 

42 

19 

473710 

675 

979855 

66 

493854 

740 

506143 

41 

20 

474115 

674 

979816 

66 

494299 

740 

505701 

40 

21 

9.474519 

674 

9.979776 

66 

9.494743 

740 

10.505257 

39 

22 

474923 

673 

979737 

66 

495186 

739 

504814 

38 

23 

475327 

672 

979697 

66 

495630 

738 

504370 

37 

24 

475730 

672 

979658 

66 

496073 

737 

503927 

36 

25 

476133 

671 

979618 

66 

496515 

737 

503485 

35 

26 

476536 

670 

979579 

66 

496957 

736 

503043 

34 

27. 

476938 

669 

979539 

66 

497399 

736 

502601 

33 

28 

477340 

669 

979499 

66 

497841 

735 

502159 

32 

29 

477741 

668 

979459 

66 

498282 

734 

501718 

31 

30 

478142 

667 

979420 

66 

498722 

734 

501278 

30 

31 

9.478542 

667 

9.979380 

66 

9.499163 

733 

10.500837 

29 

32 

478942 

666 

979340 

66 

499603 

733 

500397 

28 

33 

479312 

666 

979300 

67 

500042 

732 

499958 

27 

34 

479741 

665 

979260 

67 

500481 

731 

499519 

26 

35 

480140 

664 

979220 

67 

500920 

731 

499080 

25 

36 

480539 

663 

979180 

67 

501359 

730 

498641 

24 

37 

480937 

663 

979140 

67 

501797 

730 

498203 

A.-O 

38 

481334 

662 

979100 

67 

502235 

729 

497765 

22 

39 

481731 

661 

979059 

67 

502672 

728 

497328 

21 

40 

482128 

661 

979019 

67 

503109 

728 

496891 

20 

41 

9.482525 

660 

9.978979 

67 

9.503546 

727 

10.496454 

19 

42 

482921 

659 

978939 

67 

503982 

727 

496018 

18 

43 

483316 

659 

978898 

67 

504418 

726 

495582 

17 

44 

483712 

658 

978858 

67 

504854 

725 

495146 

16 

45 

484107 

657 

978817 

67 

505289 

725 

494711 

15 

46 

484501 

657 

978777 

67 

505724 

724 

494276 

14 

47 

484895 

656 

978736 

67 

506159 

724 

493841 

13 

48 

485289 

655 

978696 

68 

506593 

723 

493407 

12 

49 

485682 

655 

978655 

68 

507027 

722 

492973 

11 

50 

486075 

654 

978615 

68 

507480 

722 

492540 

10 

51 

9.486467 

653 

9.978574 

68 

9.507893 

"  721 

10.492107 

9 

52 

486860 

653 

978533 

68 

508326 

721 

491674 

8 

53 

487251 

652 

£78493 

68 

508759 

720 

491241 

7 

54 

487643 

651 

978452 

68 

509191 

719 

490809 

6 

55 

488034 

651 

978411 

68 

509622 

719 

490378 

5 

56 

488424 

650 

978370 

68 

510054 

718 

489946 

4 

57 

488814 

650 

978329 

68 

510485 

718 

489515 

3 

58 

489204 

649 

978288 

68 

510916 

717 

489084 

2 

59 

489593 

648 

978247 

68 

511346 

716 

488654 

1 

60 

489982 

648 

978206 

68 

5117.76 

716 

488224 

0 

Cosine 

Sine   1    1  Cotang. 

'Jang. 

M. 

72  Degrees. 


(18  Degrees.)    A  TABLE  or  LOGARITHMIC 


M. 

Sine 

D.   \  Cosine   |  1).    Tang.     D.   [   Cirtaiw. 

0 

9.4899821  648 

9.  978206'  68 

9.5117761  716 

10.488224 

60 

1 

490371 

648 

978165 

68 

512206 

716 

487794 

59 

2 

490759 

647 

978124 

68 

512635 

715 

487365 

58 

3 

491147 

646 

978083 

69 

513064 

714 

486936 

57 

4 

4915351  646 

978042 

69 

513493   714 

486507 

56 

5 

491922 

645 

978001 

69 

513921)  713 

486079 

55 

6 

492308 

644 

977959 

69 

514349   713 

485651 

54 

7 

492695 

644 

977918 

69 

514777 

712 

485223 

53 

8 

493081 

643 

977877 

69 

515204 

712 

484796 

52 

9 

493466 

642 

977835 

69 

515631 

711 

484369 

51 

10 

493851 

642 

977794 

69 

516057 

710 

483943 

50 

11 

9.494236 

641 

9.977752 

69 

9.516484 

710 

10.483516 

49 

12 

494621 

641 

977711 

69 

516910J   709 

483090 

48 

13 

495005 

640 

977669 

69 

517335 

709 

482665 

47 

14 

495388 

639 

977628 

69 

517761 

708 

482239 

46 

15 

4957721  639 

977586 

69 

518185 

708 

481815 

45 

16 

496154 

638 

977544 

70 

518610 

707 

481390 

44 

17 

496537 

637 

977503 

70 

519034 

706 

480966 

43 

18 

496919 

637 

977461 

70 

519458 

706 

480542 

42 

19 

497301 

636 

977419 

70 

519882 

705 

480118 

41 

20 

497682 

636 

977377 

70 

520305 

705 

479695 

40 

21 

9.498064 

635 

9.977335 

70 

9.520728 

704 

10.479272 

39 

22 

498444 

634 

977293 

70 

521151 

703 

478849 

38 

23 

498825 

634 

977251 

70 

521573 

703 

478427 

37 

24 

499204 

633 

977209 

70 

521995 

703 

478005 

36 

25 

499584 

632 

977167 

70 

522417 

702 

477583 

35 

26 

499963 

632 

977125 

70 

522838 

702 

477162 

34 

27 

500342 

631 

977083 

70 

523259 

701 

476741 

33 

28 

500721 

631 

977041 

70 

523680 

701 

476320 

32 

29 

501099 

630 

976999 

70 

524100 

700 

475900 

31 

30 

501476 

629 

976957 

70 

524520 

699 

475430 

30 

31 

9.501854 

629 

9.976914 

70 

9.524939 

699 

10.475061 

29 

32 

502231 

628 

976872 

71 

525359 

698 

474641 

28 

33 

502607 

628 

976830 

71 

525778 

698 

474222 

27 

34 

502984 

627 

976787 

71 

526197 

697 

473803 

26 

35 

503360 

G26 

976745 

71 

526615 

697 

473385 

25 

36 

503735 

626 

976702 

71 

527033 

696 

472967 

24 

37 

504110 

625 

976660 

71 

527451 

696 

472549 

23 

38 

504485 

625 

976617 

71 

527868 

695 

472132 

22 

39 

504860 

624 

976574 

71 

528285 

695 

471715 

21 

40 

505234 

623 

976532 

71 

528702 

694 

471298 

20 

41 

9.505608 

623 

9.976489 

71 

9.529119 

693 

0.470881 

19 

42 

505981 

622 

976446 

71 

529535 

693 

470465 

18 

43 

506S54 

622 

976404 

71 

529950 

693 

470050 

17 

44 

506727 

621 

976361 

71 

530366 

692 

469634 

16 

45 

507099 

620 

976318 

71 

530781 

691 

469219 

15 

46 

507471 

620 

976275 

71 

531196 

691 

468804 

14 

47 

507843 

619 

976232 

72 

531611 

690 

468389 

13 

48 

508214 

619 

976189 

72 

532025 

690 

467975 

12 

49 

508585 

618 

976146 

72 

532439 

689 

467561 

11 

50 

508956 

618 

976103 

72 

532853 

689 

467147 

10 

51 

9.509326 

617 

9.976060 

72 

9.533266 

688 

10.466734 

9 

52 

509696 

616 

976017 

72 

533679 

688 

466321 

8 

53 

510065 

616 

975974 

72 

534092 

687 

465908 

7 

54 

510434 

615 

975930 

72 

534504 

687 

465496 

6 

55 

510803 

615 

975887 

72 

534916 

686 

465084 

c 

56 

511172 

614 

975844 

72 

535328 

686 

464672 

4 

57 

511540 

613 

975800 

72 

535739 

685 

464261 

3 

58 

511907 

613 

975757 

72 

536150 

685 

463850 

2 

59 

612275 

612 

975714 

72 

536561 

684 

463439 

1 

60 

512642 

512 

975670 

72 

536972 

684 

463028 

0 

Cosine 

Sine   | 

Cotang.  | 

Tang. 

M. 

^1  Degrees. 


SINES  AND  TANGENTS.     (19  Degrees.) 


37 


M. 

Sine   |   D. 

Cosine 

D. 

Tang.   |   D.     Cotanp.   | 

0 

9.512642 

612 

9.975670 

V3 

9.536972 

684 

10.463028 

60 

1 

513009 

611 

975627 

73 

537382 

683 

462618 

59 

2 

513375 

611 

975583 

73 

537792 

683 

462208 

58 

3 

513741 

610 

975539 

73 

538202 

682 

461798 

57 

4 

514107 

609 

975496 

73 

538611 

682 

461389 

56 

5 

514472 

609 

975452 

73 

539020 

681 

460980 

55 

6 

514837 

608 

9  ^5408 

73 

539429 

681 

460571 

54 

7 

515202 

608 

9T5365 

73 

539837 

680 

460163 

53 

8 

515566 

607 

975321 

73 

540245 

680 

459755 

52 

9 

515930 

607 

975277 

73 

540653 

679 

459347 

51 

10 

516294 

606 

975233 

73 

541061 

679 

458939 

50 

11 

9.516657 

605 

9.975189 

73 

9.541468 

678 

10.458532 

49 

12 

517020 

605 

975145 

73 

541875 

678 

458125 

48 

13 

517382 

604 

975101 

73 

542281 

677 

457719 

47 

14 

517745 

604 

975057 

73 

542688 

677 

457312 

46 

15 

518107 

603 

975013 

73 

543094 

676 

456906 

45 

16 

518468 

603 

974969 

74 

543499 

676 

456501 

44 

17 

518829 

602 

974925 

74 

543905 

675 

456095 

43 

18 

519190 

601 

974880 

74 

544310 

675 

455690 

42 

19 

519551 

601 

974836 

74 

544715 

674 

455285 

41 

20 

519911 

600 

974792 

74 

545119 

674 

454881 

40 

21 

9.520271 

600 

9.974748 

74 

9.545524 

673 

10.454476 

39 

22 

520631 

599 

974703 

74 

545928 

673 

454072 

38 

23 

520990 

599 

974659 

74 

546331 

672 

453669 

37 

24 

521349 

598 

974614 

74   546735 

672 

453265 

36 

25 

521707 

598 

974570 

74 

547133 

671 

452862 

35 

26 

522066 

597 

974525 

74 

547540 

671 

452460 

34 

27- 

522424 

596 

974481 

74 

547943 

670 

452057 

33 

28 

522781 

596 

974436 

74 

548345 

670 

451655 

32 

29 

523138 

595 

974391 

74 

548747 

669 

451253 

31 

30 

523495 

595 

974347 

75 

549149 

669 

450851 

30 

31 

9.523852 

594 

9.974302 

75 

9.549550 

668 

10.450450 

29 

32 

524208 

594 

974257 

75 

549951 

668 

450049 

28 

33 

524564 

593 

974212 

75 

550352 

667 

449648 

27 

34 

524920 

593 

974167 

75 

550752 

667 

449248 

26 

35 

525275 

592 

974122 

75 

551152 

666 

448848 

25 

36 

525630 

591 

974077 

75 

551552 

666 

448448 

24 

37 

525984 

591 

974032 

75 

551952 

665 

448048 

23 

38 

526339 

590 

973987 

75 

552351 

665 

447649 

22 

39 

526693 

590 

973942 

75 

552750 

665 

447250 

21 

40 

527046 

589 

973897 

75 

553149 

664 

446851 

20 

41 

9.527400 

589 

9.973852 

75 

9.553548 

664 

10.446452 

19 

42 

527753 

588 

973807 

75 

553946 

663 

446054 

18 

43 

528105 

588 

973761 

75 

554344 

663 

445656 

17 

44 

528458 

587 

97.3716 

76 

554741 

662 

445259 

16 

45 

528810 

587 

973671 

76 

555139 

662 

444861 

15 

46 

529161 

586 

973625 

76 

555536 

661 

444464 

14 

47 

529513 

586 

973580 

76 

555933 

661 

444067 

13 

48 

529864 

585 

973535 

76 

556329 

660 

443671 

12 

49 

530215 

585 

973489 

76 

556725 

660 

443275 

11 

50 

030565 

584 

973444 

76 

557121 

659 

442879 

10 

51 

9.530915 

584 

9.973398 

76 

9.557517 

659 

10.442483 

9 

52 

531265 

583 

973352 

76 

557913 

659 

442087 

6 

53 

531614 

582 

973307 

76 

558308 

658 

441692 

7 

54 

531963 

582 

973261 

76 

558702 

658 

441298 

6 

55 

532312 

581 

973215 

76 

559097 

657 

440903 

• 

56 

532661 

581 

973169 

76 

559491 

657 

440509 

4 

57 

533009 

580 

973124 

76 

559885 

656 

440115 

r 
t. 

58 

533357 

580 

973078 

76 

560279 

656" 

439721 

f 

59 

533704 

579 

973032 

77 

560673 

655 

439327 

] 

60 

534052 

578 

972986 

77 

561066 

655 

438934 

0 

Cosine   |      |   Sine 

Cotang. 

Tang.   J  \L 

70  Degrees. 


38 


(20  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine      D.  |  Cosine    D. 

Tang.   j   D.     Cotang. 

0 

9.634052)  578 

9.972986  77 

9.56106 

655 

10.438934 

60 

1 

534399 

577 

972940  77 

56145 

654 

438541 

59 

2 

534745 

577 

9728941  77 

56185 

654 

438149 

58 

3 

535092 

577 

972848 

77 

562244 

653 

437756 

57 

4 

535438 

576 

972802 

77 

562636 

653 

437364 

56 

5 

535783 

576 

9727551  77 

563028 

653 

436972 

55 

6 

536129 

575 

972709 

77 

563419 

652 

436581 

54 

7 

536474 

574 

972663 

77 

56381 

652 

436189 

53 

8 

536818 

574 

972617 

77 

564-202 

651 

435798 

52 

9 

537163 

573 

972570 

77 

564592 

651 

435408 

51 

10 

537507 

573 

972524 

77   564983 

650 

435017 

50 

11 

9.537851 

572 

9.972478 

77  9.565373 

650 

10.434627 

49 

12 

538194 

572 

972431 

78   565763 

649 

434237 

48 

13 

538538 

571 

972385 

78   566153 

649 

433847 

47 

14 

538880 

571 

972338 

78   566542 

649 

433458 

46 

15 

539223 

570 

972291 

78   566932 

648 

433068 

45 

16 

539565 

570 

972245 

78   567320 

648 

432680 

44 

17 

539907 

569 

972198 

78   567709 

647 

432291 

43 

18 

540249 

569 

972151 

78   568098 

647 

431902 

42 

19 

540590 

568 

972105 

79 

568486 

646 

431514 

41 

20 

540931 

5-68 

972058 

78 

568873 

646 

431127 

40 

21 

9.541272 

567 

9.972011 

71 

9.569261 

645 

10.430739 

39 

22 

541613 

567 

971964 

73 

569648 

645 

430352 

38 

23 

541953 

566 

971917 

73 

570035 

645 

429965 

37 

24 

542293 

566 

971870 

78 

570422 

644 

429578 

36 

25 

542632 

565 

971823 

78 

570809 

644 

429191 

35 

26 

542971 

565 

971776 

rs 

571195 

643 

428805 

34 

27 

543310 

564 

971729 

79 

571581 

643 

4284-19 

33 

28 

543649 

564 

971682 

79 

571967 

642 

428033 

32 

29 

543987 

563 

971635 

79 

572352 

642 

427648 

31 

30 

544325 

563 

971588 

79 

572738 

642 

427262 

30 

31 

9.544663 

562 

9.971540 

79 

9.573123 

641 

10.426877 

29 

32 

545000 

562 

971493 

79 

573507 

641 

426493 

28 

33 

545338 

561 

971446 

79 

573892 

640 

426108 

27 

34 

545674 

561 

971398 

79 

574276 

640 

425724 

26 

35 

546011 

560 

971351 

79 

574660 

639 

425340 

25 

36 

546347 

560 

971303 

79 

575044 

639 

424956 

24 

37 

546683 

559 

971256 

79 

575427 

639 

424573 

23 

38 

547019 

559 

971208 

79 

575810 

638 

424190 

22 

39 

547354 

558 

971161 

79 

576193 

638 

423807 

21 

40 

547689 

558 

971113 

79 

576576 

637 

423424 

20 

41 

9.548024 

557 

9.971066 

80 

9.576958 

637 

T07423041 

19 

42 

548359 

557 

971018 

80 

577341 

636 

422659 

18 

43 

548693 

556 

970970 

80 

577723 

636 

422277 

17 

44 

549027 

556 

970922 

80 

578104 

636 

421896 

16 

45 

549360 

555 

970874 

80 

578486 

635 

421514 

15 

46 

549693 

555 

970827 

80 

578867 

635 

421133 

14 

47 

550026 

554 

970779 

80 

579248 

634 

420752 

13 

48 

550359 

554 

970731 

80 

579629 

634 

420371 

12 

49 

550692 

553 

970683 

80 

580009 

634 

419991 

11 

50 

551024 

553 

970635 

80 

580389 

633 

419611 

10 

51 

9.551356 

552 

9.970586 

80 

9.580769 

633 

10.419231 

9 

52 

551687 

552 

970538 

80 

581149 

632 

418851 

8 

53 

552018 

552 

970490 

80 

581528 

632 

418472 

7 

54 

552349 

551 

970442 

80 

581907 

632 

418093 

6 

55 

552680 

551 

970394 

80 

582286 

631 

417714 

5 

56 

553010 

550 

970345 

81 

582665 

631 

417335 

4 

57 

553341 

550 

970297 

81 

583043 

630 

416957 

3 

58 

553670 

549 

970249 

81 

583422 

630 

416578 

2 

59 

554000 

549 

970200 

81 

583800 

629 

416200 

1 

60 

554329 

548 

970152 

81 

584177 

629 

415823 

0 

Cosine 

Sine   1      Cotang. 

Tang.    M. 

Degrees. 


SINES   AND    TANGENTS.       (21   Degrees.) 


M.     Sine      D.      Connf   D.  f  /fang.      D.      Cntang.  | 

0 

9.554329 

548  I  9.970152 

81 

9.684177 

629 

0.415823 

60 

1 

554658 

548 

970103 

81 

584555 

629 

415445 

59 

2 

554987 

547. 

970055 

81 

584932 

628 

415068 

58 

3 

555315 

547 

970006 

81 

585309 

628 

414691 

57 

4 

555643 

546 

969957 

81 

585686 

627 

4143-14 

56 

5 

'555971 

546 

969909 

81 

586062 

627 

413938 

55 

6 

556299 

545 

969860 

81 

586439 

627 

413561 

54 

7 

556626 

545 

969811 

81 

586815 

626 

413185 

53 

8 

556953 

544 

969762 

81 

587190 

626 

412810 

52 

9 

557280 

544 

969714 

81 

587566 

625 

412434 

51 

10 

557606 

543 

969665 

81 

587941 

625 

412059 

50 

11 

9.557932 

543 

9.969616 

82 

9.588316 

625 

0,411684 

49 

12 

558258 

543 

969567 

82 

588691 

624 

411309 

48 

13 

558583 

542 

969518 

82 

589066 

624 

410934 

47 

14 

558909 

542 

969469 

82 

589440 

623 

410560 

46 

15 

559234 

541 

969420 

82 

589814 

623 

410186 

45 

16 

559558 

641 

969370 

82 

590188 

623 

409812 

44 

17 

559883 

540 

969321 

82 

590562 

622 

409438 

43 

18 

560207 

540 

969272 

82 

590935 

622 

409065 

42 

19 

560531 

539 

969223 

82 

691308 

622 

408692 

41 

20 

560855 

539 

969173 

82 

591681 

621 

408319 

40 

21 

9.561178 

538 

9.969124 

82 

9.592054 

621 

10.407946 

39 

22 

561501 

538 

969075 

82 

592426 

620 

407574 

38 

23 

561824 

537 

969025 

82 

592798 

620 

407202 

37 

24 

562146 

537 

968976 

82 

593170 

619 

406829 

36 

25 

562468 

536 

968926 

83 

593542 

619 

406458 

35 

26 

562790 

536 

968877 

83 

593914 

618 

406086 

34 

27 

563112 

536 

968827 

83 

594285 

618 

405715 

33 

28 

563433 

535 

968777 

83 

594656 

618 

405344 

32 

29 

563755 

535 

968728 

83 

595027 

617 

404973 

31 

30 

564075 

534 

968678 

83 

595398 

617 

404602 

30 

31 

9.564396 

534 

9.968628 

83 

9.595768 

617 

10.404232 

29 

32 

564716 

533 

968578 

83 

596138 

616 

403862 

28 

33 

565036 

533 

968528 

83 

596508 

616 

403492 

27 

34 

565356 

532 

968479 

83 

596878 

616 

403122 

26 

35 

565676 

532 

968429 

83 

597247 

615 

402753 

25 

36 

565995 

531 

968379 

83 

597616 

615 

402384 

24 

37 

566314 

531 

968329 

83 

597985 

615 

402015 

23 

38 

566632 

531 

968278 

83 

598354 

614 

401646 

22 

39 

566951 

530 

968228 

84 

598722 

614 

401278 

21 

40 

567269 

530 

968178 

84 

599091 

613 

400909 

20 

41 

9.567587 

529 

9.968128 

84 

9.599459 

613 

10.400541 

19 

42 

567904 

529 

968078 

84 

59982~ 

613 

400173 

18 

43 

568222 

528 

968037 

84 

600194 

612 

399806 

17 

44 

568539 

528 

967977 

84 

600562 

612 

399438 

16 

45 

568856 

528 

967927 

84 

60092 

611 

399071 

15 

46 

569172 

527 

967876 

84 

60129 

Gil 

398704 

14 

47 

569488 

527 

967826 

84 

60166 

611 

398338 

13 

48 

569804 

526 

96777 

84 

60202 

610 

397971 

12 

49 

570120 

526 

967725 

84 

60239 

610 

397605 

11 

50 

570435 

525 

967674 

84 

60276 

610 

397239 

10 

M 

9.570751 

525 

9.967624 

84 

9.60312 

609 

10.396873 

52 

571066 

524 

967573 

84 

60349 

609 

396507 

53 

57138(] 

524 

967522 

85 

60385 

609 

396142 

54 

571  69E 

523 

967471 

85 

60422 

608 

395777 

55 

57200= 

523 

967421 

85 

60458 

608 

395412 

56 

57232J 

523 

96737C 

85 

60495 

607 

395047 

57 

57263C 

522 

9673  1G 

85 

60531 

607 

394683 

58 

57295( 

522 

96726S 

85 

60568 

607 

394318 

59 

57326J 

521 

96721? 

85 

60604 

606 

393954 

60 

57357? 

521 

96716t 

85 

60641 

606 

393590 

Cosine 

Sine 

Cotang.              Tang.    M. 

Go  Decrees. 


40 


(22  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.  |   Sine      D.     Cosine    D.    Tang.     D.      Cotang.   | 

0 

9.573575 

521 

9.967166 

85 

9.606410 

606 

10.393590 

60 

1 

573888 

520 

967115 

85 

606773 

606 

393227 

59 

2 

574200 

520 

967064 

85 

607137 

605 

392863 

58 

3 

574512 

519 

967013 

85 

607500 

605 

392500 

57 

4 

574824 

519 

966961 

85 

607863 

604 

392137 

56 

5 

575136 

519 

966910 

85 

608225 

604 

391775 

55 

6 

575447 

518 

966859 

85 

608588 

604 

391412 

54 

7 

575758 

518 

966808 

85 

608950 

603 

391050 

53 

8 

576069 

517 

966756 

86 

609312 

603 

390688 

52 

9 

576379 

51T 

966705 

86 

609674 

603 

390326 

51 

10 

576689 

516 

966653 

86 

610036 

602 

389964 

50 

11 

9.576999 

516 

9.966602 

86 

9.610397 

602 

10.389603 

49 

12 

577309 

516 

966550 

86 

610759 

602 

389241 

48 

13 

577618 

515 

966499 

86 

611120 

601 

388880 

47 

14 

577927 

515 

966447 

86 

611480 

601 

388520 

46 

15 

578236 

514 

966395 

86 

611841 

601 

388159 

45 

16 

578545 

514 

966344 

86 

612201 

600 

387799 

44 

17 

578853 

513 

966292 

86 

612561 

600 

387439 

43 

18 

579162 

513 

966240 

86 

612921 

600 

387079 

42 

19 

579470 

513 

966188 

86 

613281 

599 

386719 

41 

20 

579777 

512 

966136 

86 

613641 

599 

386359 

40 

21 

9.580085 

512 

9  966085 

87 

9.614000 

598 

10.386000 

39 

22 

580392 

511 

966033 

87 

614359 

598 

385641 

38 

23 

580699 

511 

965981 

87 

6U718 

598 

385282 

37 

24 

581005 

511 

965928 

87 

615077 

597 

384923 

36 

25 

581312 

510 

965876 

87 

615435 

597 

384565 

35 

26 

581618 

510 

965824 

87 

615793 

597 

384207 

34 

27 

581924 

509 

965772 

87 

616151 

596 

383849 

33 

28 

582229 

509 

965720 

87 

616509 

596 

383491 

32 

29 

582535 

509 

965668 

87 

616867 

596 

383133 

31 

30 

582840 

508 

965615 

87 

617224 

595 

382776 

30 

31 

9.583145 

508 

9.965563 

87 

9  617582 

595 

10.382418 

29 

32 

583449 

507 

965511 

87 

617939 

595 

382061 

28 

33 

583754 

507 

965458 

87 

618295 

594 

381705 

27 

34 

584058 

506 

965406 

87 

618652 

594 

381348 

26 

35 

584361 

506 

965353 

88 

619008 

594 

380992 

25 

36 

584665 

*06 

965301 

88 

619364 

593 

380636 

24 

37 

584968 

505 

965248 

88 

619721 

593 

380279 

23 

38 

585272 

505 

965195 

88 

620076 

593 

379924 

22 

39 

585574 

504 

965143 

88 

620432 

592 

379568 

21 

40 

585877 

504 

965090 

88 

620787 

592 

379213 

20 

41 

9.586179 

503 

9.965037 

88 

9.621142 

592 

10.376858 

19 

42 

586482 

503 

964984 

88 

621497 

591 

378503 

18 

43 

586783 

503 

964931 

88 

621852 

591 

378148 

17 

44 

587085 

502 

964879 

88 

622207 

590 

377793 

16 

45 

587386 

502 

964826 

88 

622561 

590 

377439 

15 

46 

587688 

501 

964773 

88 

622915 

590 

377085 

14 

47 

587989 

501 

964719 

88 

623269 

589 

376731 

13 

48 

588289 

501 

964666 

89 

623623 

589 

376377 

12 

49 

588590 

500 

964613 

89 

623976 

589 

376024 

11 

50 

.  588890 

500 

964560 

89 

624330 

588 

375670 

10 

51 

9.589190 

499 

9.964507 

89 

9.624683 

588 

10.375317 

9 

52 

589489 

499 

964454 

89 

625036 

588 

374964 

8 

53 

589789 

499 

964400 

89 

625388 

587 

374612 

7 

54 

590088 

498 

964347 

89 

625741 

587 

374259 

6 

55 

590387 

498 

964294 

89 

626093 

587 

373907 

5 

56 

590686 

497 

964240 

89 

626445 

586 

373555 

4 

57 

590984 

497 

964187 

89 

626797 

586 

373203 

3 

58 

591282 

497 

964133 

89 

627149 

586 

372851 

2 

59 

591580 

496 

964080 

89 

627501 

585 

372499 

1 

60 

591878 

496 

964026 

89 

627852 

585 

372148 

0 

Cosine 

Sine   |    |  Cotang. 

Tang.   j  M. 

67  Degrees. 


SINES  AND  TANGENTS.      (23  Degrees.) 


41 


M.    Sine   |   D.     Cosine   |  D.  |   Tanir.     D. 

Cotang.  j 

0 

y.  591878 

496 

9  .  964026 

89 

9.627852 

585 

10.372148 

60 

1 

592176 

495 

963972 

89 

628203 

585 

371797 

59 

2 

592473 

495 

963919 

89 

628554 

585 

371446 

58 

3 

596770 

495 

963865 

90 

628905 

584 

371095 

57 

4 

593067 

494 

963811 

90 

629255 

584 

370745 

56 

5 

593363 

494    963757 

90 

629606 

583 

370394 

55 

6 

593659 

493 

963704 

90 

629956 

583 

370044 

54 

7 

593955 

493 

963650 

90 

630306 

583 

369694 

53 

8 

594251 

493 

963596  90 

630656 

583 

369344 

52 

9 

594547 

492 

963542:  90 

631005 

582 

368995 

51 

10 

594842 

492 

963488 

90 

631355 

582 

368645 

50 

11 

9.595137 

491 

9.963434 

90 

9.631704 

582 

10.368296 

49 

12 

595432 

491 

963379 

90 

632053 

581 

367947 

48 

13 

595727 

491 

963325 

90 

632401 

581 

367599 

47 

14 

596021 

490 

963271 

90 

632750 

581 

367250 

46 

II 

596315 

490 

963217 

90 

633098 

580 

366902 

45 

16 

596609 

489 

963163 

90 

633447 

580 

366553 

44 

17 

596903 

489 

963108 

91 

633795 

580 

366205 

43 

18 

597196 

489 

963054 

91 

634143 

579 

365857 

42 

19 

597490 

488 

962999 

91 

634490 

579 

365510 

41 

20 

597783 

488 

962945 

91 

634838 

579 

365162 

40 

21 

9  .  598075 

487 

9.962890 

91 

9.635185 

578 

10.364815 

39 

22 

598368 

487 

962836 

91 

635532 

578 

364468 

38 

23 

598660 

487 

962781 

'91 

635879 

578 

364121 

37 

24 

598952 

486 

962727 

91 

638226 

577 

363774 

36 

25 

599244 

486 

962672 

91 

636572 

577 

363428 

35 

26 

.  599536 

485 

962617 

91 

636919 

577 

363081 

34 

27 

599827 

485 

962562 

91 

637265 

577 

362735 

33 

28 

600118 

485 

962508 

91 

637611 

576 

362389 

32 

29 

600409 

484 

962453 

91 

637956 

576 

362044 

31 

30 

600700 

484 

962398 

92 

638302 

576 

361698 

30 

31 

9.600990 

484 

9.962343 

92 

9.638647 

575 

10.361353 

29 

32 

601280 

483 

962288 

92 

638992 

575 

361008 

28 

33 

601570 

483 

962233 

92 

639337 

575 

360663 

27 

34 

601860 

482 

962178 

92 

639682 

574 

360318 

26 

35 

602150 

482 

962123 

92 

640027 

574 

359973 

25 

36 

602439 

482 

962067 

92 

640371 

574 

359629 

24 

37 

602728 

481 

962012 

92 

640716 

573 

359284 

23 

38 

603017 

481 

961957 

92 

641060 

573 

358940 

22 

39 

603305 

481 

961902 

92 

641404 

573 

358596 

21 

40 

603594 

480 

961846 

92 

641747 

572 

358253 

20 

41 

9.603882 

480 

9.961791 

92 

9.642091 

572 

10.357909 

19 

42 

604170 

479 

961735 

92 

642434 

572 

357566 

18 

43 

604457 

479 

961680 

92 

642777 

572 

357223 

17 

44 

604745 

479 

961624 

93 

643120 

571 

356880 

16 

45 

605032 

478 

961569 

93 

643463 

571 

35653? 

15 

46 

605319 

478 

961513 

93 

643806 

571 

356194 

14 

47 

605606 

478 

961458 

93 

644148 

570 

355852 

13 

48 

605892 

477 

961402 

93 

644490 

570 

355510 

12 

49 

606179 

477 

961346 

93 

644832 

570 

355168 

11 

50 

606465 

476 

961290 

93 

645174 

569 

354826 

10 

51 

9.606751 

476 

9.961235 

93 

9.645516 

569 

107354484 

9 

52 

607036 

476 

961179 

93 

645857 

569 

354143 

8 

53 

607322 

475 

961123 

93 

646199 

569 

353801 

7 

54 

607607 

475 

961067 

93 

646540 

568 

353460 

6 

55 

607892 

474 

961011 

93 

646881 

568 

353119 

5 

56 

608177 

474 

960955 

93 

647222 

568 

352778 

4 

57 

608461 

474 

960899 

93 

647562 

567 

352438 

3 

53 

608745 

473 

960843 

94 

647903 

567 

352097 

2 

59 

609029 

473 

960786 

94 

648243 

567 

351757 

1 

60 

609313 

473 

960730 

94 

648583 

566 

351417 

0 

Cosine 

Sine 

|  Cotang.   1          Tang. 

M. 

66  Degrees. 

F 

42 


(£4  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M  |   ^ine 

D.     Cosine  |  D. 

Tanp.   1   D. 

Cotang.  | 

0 

9.609313 

473 

9.960730 

94 

9.648583 

566 

10.351417 

60 

1 

609597 

472 

960674 

94 

648923 

566 

351077 

59 

2 

609880 

472 

960618 

94 

649263 

566 

350737 

58 

3 

610164 

472 

960561 

94 

649602 

566 

350398 

57 

4 

610447 

471 

960505 

94 

649942 

565 

350058 

56 

5 

610729 

471 

960448 

94 

650281 

565 

349719 

55 

6 

611012 

470 

960392 

94 

650020 

565 

349380 

54 

7 

611294 

470 

960335 

94 

650959 

564 

349041 

53 

8 

611576 

470 

960279 

94 

651297 

564 

348703 

52 

9 

611858 

469 

960222 

94 

651636 

564 

348364 

51 

10 

612140 

469 

960165 

94 

651974 

563 

348026 

50 

11 

9.612421 

469 

9.960109 

95 

9.652312 

563 

10.347688 

49 

12 

612702 

468 

960052 

95 

652650 

563 

347350 

48 

13 

612983 

468 

959995 

95 

652988 

563 

347012 

47 

14 

613264 

467 

959938 

95 

653326 

562 

346674 

46 

15 

613545 

467 

959882 

95 

653663 

562 

346337 

45 

16 

613825 

467 

959825 

95 

654000 

562 

346000 

44 

17 

614105 

466 

959768 

95 

654337 

561 

345663 

43 

18 

614385 

466 

959711 

95 

654674 

561 

345326 

42 

19 

614665 

466 

959654 

95 

655011 

561 

344989 

41 

20 

614944 

465 

959596 

95 

655348 

561 

344652 

40 

21 

9.615223 

465 

9.959539 

95 

9.655684 

560 

10.344316 

39 

22 

615502 

465 

959482 

95 

656020 

560 

343980 

38 

23 

615781 

464 

959425 

95 

656356 

560 

343644 

37 

24 

616060 

464 

959368 

95 

656692 

559 

343308 

36 

25 

616338 

464 

959310 

96 

657028 

559 

342972 

35 

26 

616616 

463 

959253 

96 

657364 

559 

342636 

34 

27 

616894 

463 

959195 

96 

657699 

559 

342301 

33 

28 

617172 

462 

959138 

96 

658034 

558 

341966 

32 

29 

617450 

462 

959081 

96 

658369 

558 

341631 

31 

30 

617727 

462 

959023 

96 

658704 

558 

341296 

30 

31 

9.618004 

461 

9.958965 

96 

9  .  659039 

558 

10.340961 

29 

32 

618281 

461 

958908 

96 

659373 

557 

340627 

28 

33 

618558 

461 

958850 

96 

659708 

557 

340292 

27 

34 

618834 

460 

95879^ 

96 

660042 

557 

339958 

26 

35 

619110 

460 

958734 

96 

.  660376 

557 

339624 

25 

36 

619386 

460 

958677 

96 

660710 

556 

339290 

24 

37 

619662 

459 

958619 

96 

661043 

556 

338957 

23 

38 

619938 

459 

958561 

96 

661377 

556 

338623 

22 

39 

620213 

459 

958503 

97 

661710 

555 

338290 

21 

40 

620488 

458 

958445 

97 

662043 

555 

337957 

20 

41 

9.620763 

'  458 

9.958387 

97 

9.662376 

555 

10.337624 

19 

42 

621038 

457 

958329 

97 

662709 

554 

337291 

18 

43 

621313 

457 

958271 

97 

663042 

554 

336958 

17 

44 

621587 

457 

958213 

97 

663375 

554 

336625 

16 

45 

621861 

456 

958154 

97 

663707 

554 

336293 

15 

46 

622135 

456 

958096 

97 

664039 

553 

335961 

14 

47 

622409 

456 

958038 

97 

664371 

553 

335629 

13 

48 

622682 

455 

957979 

97 

664703 

553 

335297 

12 

49 

622956 

455 

957921 

97 

665035 

553 

334965 

11 

50 

623229 

455 

957863 

97 

665366 

552 

334634 

10 

51 

9.623502 

454 

9.957804 

9V 

9.665697 

552 

1^7334303 

9 

52 

623774 

454 

957746 

98 

666029 

552 

333971 

8 

53 

624047 

454 

957687 

98 

666360 

551 

333640 

7 

54 

624319 

453 

957628 

98 

666691 

551 

333309 

6 

55 

624591 

453 

957570 

98 

667021 

551 

332979 

5 

56 

624863 

453 

957511 

98 

667352 

551 

332648 

4 

57 

625135 

452 

957452 

98 

667682 

550 

332318 

3 

58 

625406 

452 

957393 

98 

668013 

550 

331987 

2 

59 

625677 

452 

957335 

98 

668343 

550 

331657 

60 

625948 

451 

957276 

98 

668672 

550 

331328 

0 

Cosine  I       I   Sine   | 

Colling 

Tang.   |  M. 

65  Degrees. 


SINES  AND  TANGENTS.     (25  Degrees.) 


M.    Sine     D.   |   Cosine  |  D.     Tang.   |   D.     Cotang.  j 

0 

9  .  625948 

451 

9.957276 

981  9.668673 

550 

10.331327|60 

1 

626219 

451 

957217 

98'   669002 

549 

330998 

59 

2 

626490 

451 

957158 

98 

669332 

549 

330668 

58 

3 

626760 

450 

957099 

98 

669661 

549 

330339 

57 

4 

627030 

450 

957040 

98 

669991 

548 

330009 

56 

5 

627300 

450 

956981 

98 

670320 

548 

329680 

55 

6 

627570 

449 

956921 

99 

670649 

548 

329351 

54 

7 

627840 

449 

956862 

99 

670977 

548 

329023 

53 

8 

628109 

449 

956803 

99 

671306 

547 

328694 

52 

9 

628378 

448 

956744 

99 

671634 

547 

328366 

51 

10 

628647 

448 

956684 

99 

.671963 

547 

328037 

50 

11 

9.628916 

447 

9.956625 

99 

9.672291 

547 

10.327709 

49 

12 

629185 

447 

956566 

99 

672619 

546 

327381 

48 

13 

629453 

447 

956506 

99 

672947 

546 

327053 

47 

14 

629721 

446 

956447 

99 

673274 

546 

326726 

46 

15 

629989 

446 

956387 

99 

673602 

546 

326398 

45 

16 

630257 

446 

956327 

99 

673929 

545 

326071 

44 

17 

630524 

446 

956268 

99 

674257 

545 

325743 

43 

18 

630792 

445 

956208 

100 

674584 

545 

325416 

42 

19 

631059 

445 

956148 

100 

674910 

544 

325090 

41 

20 

631326 

445 

956089 

100 

675237 

544 

324763 

40 

21 

9.631593 

444 

9.956029 

100 

9.675564 

544 

10.324436 

39 

22 

631859 

444 

955969 

100 

675890 

544 

324110 

38 

23 

632125 

444 

955909 

100 

676216 

543 

323784 

37 

24 

632392 

443 

955849 

100 

676543 

543 

323457 

36 

25 

632658 

443 

955789 

100 

676869 

543 

323131 

35 

26 

.  632923 

443 

955729 

100 

677194 

543 

322806 

34 

27 

633189 

442 

955669 

100 

677520 

542 

322480 

33 

28 

633454 

442 

955609 

100 

677846 

542 

322154 

32 

29 

633719 

442 

955548 

100 

678171 

542 

321829 

31 

30 

633984 

441 

955488 

100 

678496 

542 

321504 

30 

31 

9.634249 

441 

9.955428 

101 

9.678821 

541 

10.321179 

29 

32 

634514 

440 

955368 

101 

679146 

541 

320854 

28 

33 

634778 

440 

955307 

101 

679471 

541 

320529 

27 

34 

635042 

440 

955247 

101 

679795 

541 

320205 

26 

35 

635306 

439 

955186 

101 

680120 

540 

319880 

25 

36 

635570 

439 

955126 

101 

680444 

540 

319556 

24 

37 

635834 

439 

955065 

101 

680768 

540 

319232 

23 

38 

636097 

438 

955005 

101 

681092 

540 

318908 

22 

39 

636360 

438 

954944 

101 

681416 

539 

318584 

21 

40 

636623 

438 

954883 

101 

681740 

5C° 

318260 

20 

41 

9.636886 

437 

9  954823 

101 

9.682063 

539 

10.317937 

19 

42 

637148 

437 

954762 

101 

682387 

539 

317613 

18 

43 

63741  1 

437 

954701 

101 

682710 

538 

317290 

17 

44 

637673 

437 

954640 

101 

683033 

538 

316967 

16 

45 

637935 

436 

954579 

101 

683356 

538 

316644 

15 

46 

638197 

436 

954518 

102 

683679 

538 

316321 

14 

47 

638458 

436 

954457 

102 

684001 

537 

315999 

13 

48 

638720 

435 

954396 

102 

684324 

537 

315676 

12 

49 

638981 

435 

954335 

102 

684646 

537 

315354 

11 

50 

639242 

435 

954274 

102 

684968 

537 

315032 

10 

51 

9  .  639503 

434 

9.954213 

102 

9.685290 

536 

10.314710 

9 

52 

639764 

434 

954152 

102 

685612 

536 

314388 

8 

53 

640024 

434 

954090 

102 

685934 

536 

314066 

7 

54 

640284 

433 

954029 

102 

686255 

536 

313745 

6 

55 

640544 

433 

953968 

102 

686577 

C35 

313423 

5 

56 

640804 

433 

953906 

102 

686898 

535 

313102 

4 

57 

641064 

432 

953845 

102 

687219 

535 

812781 

3 

58 

641324 

432 

953783 

102 

687540 

535 

312460 

t 

59 

641584 

432 

953722 

103 

687861 

534 

31-2139 

] 

60 

641842 

431 

953660 

103   688182 

534 

311818 

0 

Cosine 

|   Sine 

(  Cotang. 

Tang.   |  M. 

64  Degrees. 


44 


(26  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine      D.    '  Cosine   |  1).  |   T-nr.   1   D      Cutaii*. 

0 

9.641842 

431 

9.953660 

103 

9.688182 

534 

10.311818 

60 

1 

642101 

431 

953599 

103 

688502 

534 

311498 

59 

2 

642360 

431 

953537 

103 

688823 

534 

311177 

58 

3 

642618 

430 

953475 

103 

689143 

533 

310857 

57 

4 

642877 

430 

953413 

103 

689463 

533 

310537 

56 

5 

643135 

430 

953352 

103 

689783 

533 

310217 

55 

6 

»  643393 

430 

953290 

103 

690103 

533 

309897 

54 

7 

643650 

429 

953228 

103 

690423 

533 

309577 

53 

8 

643908 

429 

953166 

103 

690742 

532 

309258 

52 

9 

644165 

429 

953104 

103 

691062 

532 

308938 

51 

10 

644423 

428 

953042 

103 

691381 

532 

308619 

50 

11 

9  .  644680 

428 

9.952980 

104 

9.691700 

531 

10.308300 

49 

12 

644936 

428 

952918 

104 

692019 

531 

307981 

48 

13 

645193 

427 

952855 

104 

692338 

531 

307662 

47 

14 

645450 

427 

952793 

104 

692656 

531 

307344 

46 

15 

645706 

427 

952731 

104 

692975 

531 

307025 

45 

16 

645962 

426 

952669 

104 

693293 

530 

306707 

44 

17 

646218 

426 

952606 

104 

693612 

530 

306388 

43 

18 

646474 

426 

952544 

104 

693930 

530 

306070 

42 

19 

646729 

425 

952481 

104 

694248 

530 

305752 

41 

20 

646984 

425 

952419 

104 

694566 

529 

305434 

40 

21 

9.647240 

425 

9.952356 

104 

9.694883 

529 

10.3D5117 

39 

22 

647494 

424 

952294 

104 

695201 

529 

304799 

38 

23 

647749 

424 

952231 

104 

695518 

529 

304482 

37 

24 

648004 

424 

952168 

105 

695836 

529 

304164 

36 

25 

648258 

424 

952106 

105 

696153 

528 

303847 

35 

26 

648512 

423 

952043 

105 

696470 

528 

303530 

34 

27 

648766 

423 

951980 

105 

696787 

528 

303213 

33 

28 

649020 

423 

951917 

105 

697103 

528 

302897 

32 

29 

64J9274 

422 

951854 

105 

697420 

527 

302580 

31 

30 

649527 

422 

951791 

105 

697736 

527 

302264 

.30 

31 

9.649781 

422 

9.951728 

105 

JFT698053 

527 

10.301947 

29 

32 

650034 

422 

951665 

105 

698369 

527 

301631 

28 

33 

650287 

421 

951602 

105 

698685 

526 

301315 

27 

34 

650539 

421 

951539 

105 

699001 

526 

300999 

26 

35 

650792 

421 

951476 

105 

699316 

526 

300684 

25 

36 

651044 

420 

951412 

105 

699632 

526 

300368 

24 

37 

651297 

420 

951349 

106 

699947 

526 

300053 

23 

38 

651549 

420 

951286 

106 

700263 

525 

299737 

22 

39 

651800 

419 

951222 

106 

700578 

525 

299422 

21 

40 

652052 

419 

951159 

106 

700893)  525 

299107 

20 

41 

9.652304 

419 

9.951096 

106:9.701208 

524 

10.298792 

19 

42 

652555 

418 

951032 

106 

701523 

524 

298477 

18 

43 

652806 

418 

950968 

106 

701837 

524 

298163 

17 

44 

653057 

418 

950905 

106 

702152 

524 

297848 

16 

45 

653308 

418 

950841 

106 

702466 

524 

297534 

15 

46 

653558 

417 

950778 

106 

702780 

523 

297220 

14 

47 

653808 

417 

950714 

106 

703095 

523 

296905 

13 

48 

654059 

417 

950650 

106 

703409 

523 

296591 

12 

49 

654309 

416 

950586 

106 

703723 

523 

296277 

11 

50 

654558 

416 

950522 

107 

704036 

522 

295964 

10 

51 

9.654808 

416 

9.950458 

107 

9.704350 

522 

10.295650 

9 

52 

655058 

416 

950394 

107 

704663 

522 

295337 

8 

53 

655307 

415 

950330 

107 

704977 

522 

295023 

7 

54 

655556 

415 

'  950266 

107 

705290 

522 

294710 

6 

55 

655805 

415 

950202 

107 

705603 

521 

294397 

5 

56 

656054 

414 

950138 

107 

705916 

521 

294084 

4 

57 

656302 

414 

950074 

107 

706228 

521 

293772 

3 

58 

656551 

414 

950010 

107 

706511 

521 

293459 

2 

59 

656799 

413 

940945 

107 

706854 

521 

293146 

1 

60 

657047 

413 

949881 

107 

707166 

520 

292834 

0 

Cosine  j 

Sine 

|   Cotan?.  |- 

Tang.    j  Al. 

63  Degrees. 


SINES  AND  TANGENTS.     (27  Degrees.) 


45 


M. 

Sine      D.   |  Cosine    D.    Tang.   !   D.     Cotang. 

0 

«.  657047 

413 

9.949881 

107 

9,707166 

520 

10.292834 

60 

1 

657295 

413 

949816 

107 

707478 

520 

292522 

59 

2 

657542 

412 

949752 

107 

707790 

520 

292210 

58 

3 

657790 

412 

949688 

108 

708102 

520 

291898 

57 

4 

658037 

412 

949623 

108 

708414 

519 

291586 

56 

5 

658284 

412 

949558 

108 

708726 

519 

291274 

55 

6 

658531 

411 

949494 

108 

709037 

519 

290963 

54 

7 

658778 

411 

949429 

108 

709349 

519 

290651 

53 

8 

659025 

411 

949364 

108 

709660 

519 

290340 

52 

9 

659271 

410 

949300 

108 

709971 

518 

290029 

51 

10 

659517 

410 

949235 

108 

710282 

518 

289718 

50 

11 

9.659763 

410 

9.949170 

108 

9.710593 

518 

10.289407 

49 

12 

660009 

409 

949105 

108 

710904 

518 

289096 

48 

13 

660255 

409 

949040 

108 

711215 

518 

288785 

47 

14 

660501 

409 

948975 

108 

711525 

517 

288475 

46 

15 

660746 

409 

948910 

108 

711836 

517 

288164 

45 

16 

660991 

408 

948845 

108 

712146 

517 

287854 

44 

17 

661236 

408 

948780 

109 

712456 

517 

287544 

43 

18 

661481 

408 

948715 

109 

712766 

516 

287234 

42 

19 

661726 

407 

948650 

109 

713076 

516 

286924 

41 

20 

661970 

407 

948584 

109 

713386 

516 

286614 

40 

21 

9.662214 

407 

9.948519 

109 

9.713696 

516 

10.286304 

39 

22 

662459 

407 

948454 

109 

714005 

516 

285995 

38 

23 

662703 

406 

948388 

109 

714314 

515 

285686 

37 

24 

662946 

406 

948323 

109 

714624 

515 

285376 

36 

25 

663190 

406 

948257 

109 

714933 

515 

285067 

35 

26 

663433 

405 

948192 

109 

715242 

515 

284758 

34 

27 

663677 

405 

948126 

109 

715551 

514 

284449 

33 

28 

663920 

405 

948060 

109 

715860 

514 

284140 

32 

29 

664163 

405 

947995 

110 

716168 

514 

283832 

31 

30 

664406 

404 

947929 

110 

716477 

514 

283523 

30 

31 

9.664648 

404 

9.947863 

no 

9.716785 

514 

10.283215 

29 

32 

664891 

404 

947797 

110 

717093 

513 

282907 

28 

33 

665133 

403 

947731 

110 

717401 

513 

282599 

27 

34 

665375 

403 

947665 

110 

717709 

513 

282291 

26 

35 

665617 

403 

947600 

110 

718017 

513 

281983 

25 

36 

665859 

402 

947533 

110 

718325 

513 

281675 

24 

37 

666100 

402 

947467 

110 

718633 

512 

281367 

23 

38 

666342 

402 

947401 

110 

718940 

512 

281060 

22 

39 

666583 

402 

947335 

110 

719248 

512 

280752 

21 

40 

666824 

401 

947269 

110 

719555 

512 

280445 

20 

41 

9.667065 

401 

9.947203 

110 

9.719862 

512 

10.280138 

19 

42 

667305 

401 

947136 

111 

720169 

511 

279831 

18 

43 

667546 

401 

947070 

111 

720476 

511 

279524 

17 

44 

667786 

400 

947004 

111 

720783 

511 

279217 

16 

45 

668027 

400 

946937 

111 

721089 

511 

278911 

15 

46 

668267 

400 

946871 

111 

721396 

511 

278604 

14 

47 

668506 

399 

946804 

111 

721702 

510 

278298 

13 

48 

668746 

399 

946738 

111 

722009 

510 

277991 

12 

49 

668986 

399 

946671 

111 

722315 

510 

277685 

11 

50 

669225 

399 

946604 

111 

722621 

510 

277379 

10 

51 

9.669464 

398 

9.946538 

111 

9.722927 

510 

10.277073 

9 

52 

669703 

398 

946471 

111 

723232 

509 

276768 

8 

53 

669942 

398 

946404 

111 

723538 

509 

276462 

7 

54 

670181 

397 

946337 

111 

723844 

509 

276156 

6 

55 

670419 

397 

946270 

112 

724149 

509 

275851 

5 

56 

670658 

397 

946203 

112 

724454 

509 

275546 

4 

57 

670896 

397 

946136 

112 

724759 

508 

275241 

a 

58 

671134 

396 

946069 

112 

725065 

508 

274935 

2 

59 

671372 

396 

946002 

112 

725369 

508 

274631 

1 

60 

671609 

396 

945935 

112 

725674 

508 

274326 

0 

j  Cosine 

Sine 

|  Cotang. 

Tang.  |  M. 

Degrees. 


•*.* 


(28  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine 

D.     Cosine 

D 

T.IMK.     1).     Coranat.  | 

0 

9.67160 

396 

9.94593.r 

m 

9.72567 

508 

10.274321 

60 

1 

67184 

395 

945866 

m 

72597 

508 

274021 

59 

2 

67208 

395 

94580C 

112 

72628 

507 

2737H 

58 

3 

67232 

395 

94573? 

112 

72658 

507 

273412 

57 

4 

67255 

395 

94566fi 

112 

72689 

507 

2731  OS 

56 

5 

67279 

394 

94559$ 

112 

72719 

507 

272803 

55 

6 

67303 

394 

945531 

112 

72750 

507 

272499 

54 

7 

67326 

394 

945464 

113 

72780 

506 

272195 

53 

8 

67350 

394 

945396 

113 

72810 

506 

271891 

52 

9 

67374 

393 

945328 

113 

72841 

506 

271588 

51 

10 

67397 

393 

945261 

113 

72871 

506 

271284 

50 

11 

9.67421 

393 

9.945193 

113 

9.729020 

506 

10.270980 

49 

12 

674448 

392 

945125 

113 

729323 

505 

270677 

48 

13 

674684 

392 

945058 

113 

729626 

505 

270374 

47 

14 

674919 

392 

944990 

113 

729929 

505 

270071 

46 

15 

675155 

392 

944922 

113 

730233 

505 

269767 

45 

16 

675390 

391 

944854 

113 

730535 

505 

269465 

44 

17 

675624 

391 

944786 

113 

730838 

504 

269162 

43 

18 

675859 

391 

944718 

113 

731141 

504 

268859 

42 

19 

676094 

391 

944650 

113 

731444 

504 

268556 

41 

20 

676328 

390 

944582 

114 

731746 

504 

268254 

40 

21 

9.676562 

390 

9.944514 

114 

9.732048 

504 

10.267952 

39 

22 

676796 

390 

944446 

114 

732351 

503 

267649 

38 

23 

677030 

390 

944377 

114 

732653 

503 

267347 

37 

24 

677264 

389 

944309 

114 

732955 

503 

267045 

36 

25 

677498 

389 

944241 

114 

733257 

503 

266743 

35 

26 

677731 

389 

944]  72 

114 

733558 

503 

266442 

34 

27 

677964 

388 

944104 

114 

733860 

502 

266140 

33 

28 

678197 

388 

944036 

114 

734162 

502 

265838 

32 

29 

678430 

388 

943967 

114 

734463 

502 

265537 

31 

30 

678683 

388 

943899 

114 

734764 

502 

265236 

30 

31 

9  678895 

387 

9.943830 

114 

9.735066 

502 

10.264931 

29 

32 

679128 

387 

943761 

114 

735367 

502 

264633 

28 

33 

679360 

387 

943693 

115 

735668 

501 

264332 

27 

34 

679592 

387 

943624 

115 

735969 

501 

264031 

26 

35 

679824 

386 

943555 

115 

736269 

501 

263731 

25 

36 

680056 

386 

943486 

115 

736570 

501 

263430 

24 

37 

680288 

386 

943417 

115 

736871 

501 

263129 

23 

38 

680519 

385 

943348 

115 

737171 

500 

262829 

22 

39 

680750 

385 

943279 

115 

737471 

500 

262529 

21 

40 

680982 

385 

943210 

115 

737771 

500 

262229 

20 

41 

9.681213 

385 

9.943141 

115 

9.738071 

500 

10.261929 

19 

42 

681443 

384 

943072 

115 

738371 

500 

261629 

18 

43 

681674 

384 

943003 

115 

738671 

499 

261329 

17 

44 

681905 

384 

942934 

115 

738971 

499 

261029 

16 

45 

682135 

384 

942864 

115 

739271 

499 

260729 

15 

46 

682365 

383, 

942795!  116 

739570 

499 

260430 

14 

47 

"682595 

383 

942726 

116 

739870 

499 

260130 

13 

48 

682825 

383 

942656 

116 

740169 

499 

259831 

12 

49 

683055 

383 

942587 

116 

740468 

498 

259532 

11 

50 

683284 

382 

942517 

116 

740767 

498 

259233 

10 

61 

9.683514 

382 

9  .  942448 

116 

9.741066 

498 

10.258934 

9 

c.2 

683743 

382 

942378 

116 

741365 

498 

258635 

8 

53 

683972 

382 

942308 

116 

741664 

498 

258336 

7 

54 

684201 

381 

942239 

116 

741962 

497 

258038 

6 

55 

684430 

381 

942169 

116 

742261 

497 

257739 

5 

56 

684658 

381 

942099 

116 

742559 

497 

257441 

4 

57 

684887 

380 

942029 

116 

742858 

497 

257142 

3 

58 

685115 

380 

941959 

116 

743156 

497 

256844 

2 

59 

685343 

380 

941889 

117 

743454 

497 

256546 

1 

60  ' 

6S5571 

380 

941819 

117 

743752 

496 

256248 

0 

Cosine 

Sinr:   |    |  Cotang. 

Tang.    M. 

61  Degrees. 


SINES    AM)   TANGENTS.        ^29  DegfCCS.) 


M.    Sine      D.    Cosine 

D.  |   Tang.     D.     Cotang. 

0 

9.685571 

380 

9.941819 

117 

9.743752 

496 

10.256248 

60 

1 

685799 

379 

941749 

117 

744050 

496 

255950 

59 

2 

686027 

379 

941679 

117 

744348 

496 

255652 

58 

3 

686254 

379 

941609 

117 

744645 

496 

255355 

57 

4 

686482 

379 

941539 

117 

744943 

496 

255057 

56 

5 

686709 

378 

941469 

117 

745240 

496 

254760 

55 

6 

686936 

378 

941398 

117 

745538 

495 

254462 

54 

7 

687163 

378 

941328 

117 

745835 

495 

254165 

53 

8 

687389 

378 

941258 

117 

746132 

495 

253868 

52 

9 

687616 

377 

941187 

117 

746429 

495 

253571 

51 

10 

687843 

377 

941117 

117 

746726 

495 

253274 

50 

11 

9.688069 

377 

9.941046 

118 

9.747023 

494 

10.252977 

49 

12 

688295 

377 

940975 

118 

747319 

494 

252681 

48 

13 

688521 

376 

940905 

118 

747616 

494 

252384 

47 

14 

688747 

376 

940834 

118 

747913 

494 

252087 

46 

15 

688972 

376 

940763 

118 

748209 

494 

251791 

45 

16 

689198 

376 

940693 

118 

748505 

493 

251495 

44 

17 

689423 

375 

940622 

118 

748801 

493 

251199 

43 

18 

689648 

375 

940551 

118 

749097 

493 

250903 

42 

19 

689873 

375 

940480 

118 

749393 

493 

250607 

41 

20 

%690098 

375 

940409 

118 

749689 

493 

250311 

40 

21 

9.690323 

374 

9.940338 

118 

9.749985 

493 

10.250015 

39 

22 

690548 

374 

940267 

118 

750281 

492 

249719 

38 

23 

690772 

374 

940196 

118 

750576 

492 

249424 

37 

24 

690996 

374 

940125 

119 

750872 

492 

249128 

36 

25 

691220 

373 

940054 

119 

751167 

492 

248833 

35 

26 

691444 

373 

939982 

119 

751462 

492 

248538 

34 

27 

691668 

373 

939911 

119 

751757 

492 

248243 

33 

28 

691892 

373 

939840 

119 

752052 

491 

247948 

32 

29 

692115 

372 

939768 

119 

752347 

491 

247653 

31 

30 

6923391  372 

939697 

119 

752642 

491 

247358 

,30 

31 

9.692562 

372 

9.939625 

119 

9.752937 

491 

107247063 

29 

32 

692785 

371 

939554 

119 

753231 

491 

246769 

28 

33 

693008 

371 

939482 

119 

753526 

491 

246474 

27 

34 

693231 

371 

939410 

119 

753820 

490 

246180 

26 

35 

693453 

371 

939339 

119 

754115 

490 

245885 

25 

36 

693676   370 

939267 

120 

754409 

490 

245591 

24 

37 

693898   370 

939195 

120 

754703 

490 

245297 

23 

38 

694120   370 

939123 

120 

754997 

490 

245003 

22 

39 

694342   370 

939052 

120 

755291 

490 

244709 

21 

40 

694564'  369 

938980 

120 

755585 

489 

244415 

20 

41 

9.694786 

369 

9.938908 

120 

9.755878 

489 

10.244122 

19 

42 

695007 

369 

938836 

120 

756172 

489 

243828 

18 

43 

695229 

369 

938763 

120 

756465 

489 

243535 

17 

44 

695450 

368 

938691 

120 

756759 

489 

243241 

16 

45 

695671 

368 

938619 

120 

757052 

489 

242948 

15 

46 

695892 

368 

938547 

120 

757345 

488 

242655 

14 

47 

696113 

368 

938475 

120 

757638 

488 

242362 

13 

48 

696334   367 

'  938402 

121 

757931 

488 

242069 

12 

49 

696554   367 

938330 

121 

758224 

488 

241776 

11 

50 

696775;  367 

938258 

121 

758517 

488 

241483 

10 

51 

9.696995:  367 

9.938185 

121 

9.758810 

488 

10.241190 

Q 

52 

697215 

366 

938113 

121 

759102 

487 

240898 

8 

53 

697435 

366 

938040J  121 

759395 

487 

240605 

7 

54 

697654   366 

9379671  121 

759687 

487 

240313 

6 

55 

697874J  366 

937895  121 

759979 

487 

24002 

5 

56 

6980941  365 

937822'  121 

760272 

487 

239728 

4 

57 

698313:  365 

937749  121 

760564 

487 

239436 

3 

58 

698532   355 

937670  121 

760856 

486 

239144 

2 

59 

698751 

365 

937604J  121 

761148 

486 

238852 

1 

60 

698970   364 

9375311  121 

761439 

486 

23856 

0 

|  Cosine            Sine 

Cotang.            Tang. 

M. 

Decree*. 


48 


(30  Degrees.)     A  TABLE  OP  LOGARITHMIC 


M. 

Sine 

D.   |   Cosine   |  D.  j   Tans;. 

D. 

Cotang.   | 

0 

9.698970 

364 

9.937531 

121 

9.761439 

486 

10.238561 

60 

1 

699189 

364 

937458 

122 

761731 

486 

238269 

59 

2 

699407 

364 

937385 

122 

762023 

486 

237977 

58 

3 

699626 

364 

937312 

122 

762314 

486 

237686 

57 

4 

699844 

363 

937238 

122 

762606 

485 

237394 

56 

5 

700062 

363 

937165 

122 

762897 

485 

237103 

55 

6 

700280 

363 

937092 

122 

763188 

485 

236812 

54 

7 

700498 

363 

937019 

122 

763479 

485 

236521 

53 

8 

700716 

363 

936946 

122 

763770 

485 

236230 

52 

9 

700933 

362 

936872 

122 

764061 

485 

235939 

51 

10 

701151 

362 

936799 

122 

764352 

484 

235648 

50 

11 

9.701368 

362 

9.936725 

122 

9.764643 

484 

10.235357 

49 

12 

701585 

362 

936652 

123 

764933 

484 

235067 

48 

13 

701802 

361 

936578 

123 

765224 

484 

234776 

47 

14 

702019 

361 

936505 

123 

765514 

484 

234486 

46 

15 

702236 

361 

936431 

123 

765805 

484 

234195 

45 

16 

702452 

361 

936357 

123 

766095 

484 

233905 

44 

17 

702669 

360 

936284 

123 

766385 

483 

233615 

43 

18 

702885 

360 

936210 

123 

766675 

483 

233325 

42 

19 

703101 

360 

936136 

123 

766965 

483 

233035 

41 

20 

703317 

360' 

93M62 

123 

767255 

483 

232745 

40 

21 

9.703533 

359 

9  .  935988 

123 

9.767545 

483 

10.232455 

39 

22 

703749 

359 

935914 

123 

767834 

483 

232166 

38 

23 

703964 

359 

935840 

123 

768124 

482 

231876 

37 

24 

704179 

359 

935766 

124 

768413 

482 

231587 

36 

25 

704395 

359 

935692 

124 

768703 

482 

231297 

35 

26 

704610 

358 

935618 

124 

768992 

482 

231008 

34 

27 

704825 

358 

935543 

124 

769281 

482 

230719 

33 

28 

705040 

358 

935469 

124 

769570 

482 

230430 

32 

29 

705254 

358 

935395 

124 

769860 

481 

230140 

31 

30 

705469 

357 

935320 

124 

770148 

481 

229852 

30 

31 

9.705683 

357 

9.935246 

124 

9.770437 

481 

10.229563 

29 

32 

705898 

357 

935171 

124 

770726 

481 

229274 

28 

33 

706112 

357 

935097 

124 

771015 

481 

228985 

27 

34 

706326 

356 

935022 

124 

771303 

481 

228697 

26 

35 

706539 

356 

934948 

124 

771592 

481 

228408 

25 

36 

706753 

356 

934873 

124 

771880 

480 

228120 

24 

37 

706967 

356 

934798 

125 

772168 

480 

227832 

23 

38 

707180 

355 

934723 

125 

772457 

480 

227543 

22 

39 

707393 

355 

934649 

125 

772745 

480 

227255 

21 

40 

707606 

355 

934574 

125 

773033 

480 

226967 

20 

41 

9.707819 

355 

9.934499 

125 

9.773321 

480 

10.226679 

19 

42 

708032 

354 

934424 

125 

773608 

479 

226392 

18 

43 

708245 

354 

934349 

125 

773896 

479 

226104 

17 

44 

708458 

354 

934274 

125 

774184 

479 

225816 

16 

45 

708670 

354 

934199 

125 

774471 

479 

225529 

15 

46 

708882 

353 

934123 

125 

774759 

479 

225241 

14 

47 

709094 

353 

934048 

125 

775046 

479 

224954 

13 

48 

709306 

353 

933973 

125 

775333 

479 

224667 

12 

49 

709518 

353 

933898 

126 

775621 

478 

224379 

11 

50 

709730 

353 

933822 

126 

775908 

478 

224092 

10 

51 

9.709941 

352 

9.933747 

126 

9.776195 

478 

10.223805 

9 

52 

710153 

352 

933671 

126 

776482 

478 

223518 

8 

53 

710364 

352 

933596 

126 

776769 

478 

223231 

7 

54 

710575 

352 

933520 

126 

777055 

478 

222945 

6 

55 

710786 

351 

933445 

126 

777342 

478 

222658 

5 

56 

71099? 

351 

933369 

126 

777628 

477 

222372 

4 

57 

711208 

351 

933293 

126 

777915 

477 

222085 

3 

58 

711419 

351 

933217 

126 

778201 

477 

221799 

2 

59 

711629 

350 

933141 

126 

778487 

477 

221512 

1 

60 

711839 

350 

933066 

126 

778774 

477 

221226 

0 

Cosine  1 

Bine 

|  Uotaiig.   | 

Tung.   |  M. 

59  Degrees. 


SINES  AND  TANGENTS.      (31  Degrees.) 


49 


If. 

Sine 

n. 

Cosine   |  D.     Tantr.     D.     Cotang. 

0 

9  711839 

350 

9.933066 

126 

9.778774 

477 

10.221226 

60 

1 

712050 

350 

932990 

127 

779060 

477 

220940 

59 

o 

712260 

350 

932914 

127 

779346 

176 

220654 

58 

3 

712469 

349 

932838 

127 

779632 

476 

220368 

57 

4 

712679 

849 

932762 

127 

779918 

476 

220082 

56 

5 

712889 

349 

932685 

127 

780203   476 

219797 

55 

6 

713098 

349 

932609 

127 

780489   476 

219511 

54 

7 

713308 

349 

932533 

127 

780775J  476 

219225 

53 

8 

713517 

348 

932457 

127 

781060   476 

218940 

52 

9 

713726 

348 

932380 

127 

781346 

475 

218654 

51 

10 

713935 

348 

932304 

127 

781631 

475 

218369 

50 

11 

9.714144 

348 

9.932228 

127 

9.781916   475 

10.218084 

49 

12 

714352 

347 

932151 

127 

782201 

475 

217799 

48 

13 

714561 

347 

932075 

128 

782486 

475 

217514 

47 

14 

714769 

347 

931998 

128 

782771 

475 

217229 

46 

15 

"714978 

347 

931921 

128 

783056 

475 

216944 

45 

16 

715186 

347 

931845 

128 

783341 

475 

216659 

44 

17 

715394 

346 

931768 

128 

783626 

474 

216374 

43 

18 

715602 

346 

931691 

128 

783910   474 

216090 

-42 

19 

715809 

346 

931614 

128 

784195 

474 

215805 

41 

20 

716017 

346 

931537 

128 

784479 

474 

215521 

40 

21 

9,716224 

345 

9.931460 

128 

9.784764 

474 

10.215236 

39 

22 

716432 

345 

931383 

128 

785048 

474 

214952 

38 

23 

716639 

345 

931306 

128 

785332 

473 

214668 

37 

24 

716846 

345 

931229 

129 

785616 

473 

214384 

36 

25 

717053 

345 

931152 

129 

785900 

473 

214100 

35 

26 

717259 

344 

931075 

129 

786184 

473 

213S1G 

34 

27 

717466 

344 

980998 

129 

786468 

473 

213532 

33 

28 

717673 

344 

930921 

129 

786752 

473 

213248 

32 

29 

717879 

344 

930843 

129 

787036 

473 

212964 

31 

30 

718085 

343 

930766 

129 

787319 

472 

212681 

30 

31 

9.718291 

343 

9.930688 

129 

9.787603 

472 

1(K2  12397 

29 

32 

718497 

343 

930611 

129 

787886 

472 

212114 

28 

33 

718703 

343 

930533 

129 

788170 

472 

211830 

27 

34 

718909 

343 

930456 

129 

788453 

472 

211547 

26 

35 

719114 

342 

930378 

129 

788736 

472 

211264 

25 

36 

719320 

342 

930300 

130 

789019 

472 

210981 

24 

37 

719525 

-  342 

930223 

130 

789302 

471 

210698 

23 

38 

719730 

342 

930145 

130 

789585   471 

210415 

22 

39 

719935 

341 

930067 

130 

789868 

471 

210132 

21 

40 

720140 

341 

929989 

130 

'790151 

471 

209849 

20 

41 

9.720345 

341 

9.929911 

130 

9  .  790433 

471 

10.209567 

19 

42 

720549 

341 

929833 

130 

790716 

471 

209284 

18 

43 

720754 

340 

929755 

130 

790999 

471 

209001 

17 

44 

720958 

340 

929677 

130 

791281 

471 

208719 

16 

45 

721162 

340 

929599 

130 

791563 

470 

208437 

15 

46 

721366 

340 

929521 

130 

791846 

470 

208154 

14 

47 

721570 

340 

929442 

130 

7921  28  j  470 

207872 

13 

48 

721774 

339 

929364 

131 

792410 

470 

207590 

12 

49 

721978 

339 

929286 

131 

792692 

470 

207308 

11 

50 

722181  j  339 

929207 

131 

792974 

470. 

207026 

10 

51 

9.7223851  339 

9.929129 

131 

9.793256   470 

10.206744 

9 

52 

7225881  339 

929050 

131 

793538   469 

206462 

8 

53 

722791 

338 

928972 

.131 

793819 

469 

206181 

7 

54 

722994 

338 

928893 

131 

794101 

469 

205899 

6 

55 

723197 

338 

928815 

131 

794383   469 

205617 

5 

56 

723400 

338 

928736 

131 

794664'  469 

205336 

4 

57 

723603 

337 

928657 

131 

794945   469 

205055 

3 

58 

723805 

337 

928578 

131 

795227   469 

204773 

2 

59 

724007 

337 

928499 

131 

795508   468 

204492 

1 

60 

724210 

337 

928420 

131 

795789;  468 

204211 

0 

Cosine 

Sine   | 

Cotarig. 

Tang.     M. 

58  Degrees. 

G 

50 


(32  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine      D.     Cosine   |  D. 

Tanjr. 

D. 

Ootang.  | 

0 

9.724210 

337 

9.928420 

132 

9  .  795789 

468 

10.204211 

60 

1 

724412 

337 

928342 

132 

796070 

468 

203930 

59 

2 

724614 

336 

928263 

132 

796351 

468 

203649 

58 

3 

724816 

336 

928183 

132 

796632 

468 

203368 

57 

4 

725017 

336 

928104 

132 

796913 

468 

203087 

56 

5 

725219 

336 

928025 

132 

797194 

468 

202806 

55 

6 

725420 

335 

927946 

132 

797475 

468 

202525 

54 

7 

725622 

335 

927867 

132 

797755 

468 

202245 

53 

8 

725823 

335 

927787 

132 

798036 

467 

201964 

52 

9 

726024 

335 

927708 

132 

798316 

467 

201684 

51 

10 

726225 

335 

927629 

132 

798596 

467 

201404 

50 

11 

9  .  726426 

334 

9.927549 

132 

9  .  798877 

467 

10.201123 

49 

12 

726626 

334 

927470 

133 

799157 

467 

200843 

48 

13 

726827 

334 

927390 

133 

799437 

467 

200563 

47 

14 

727027 

334 

927310 

133 

799717 

467 

200283 

46 

15 

727228 

334 

927231 

133 

799997 

466 

200003 

45 

16 

727428 

333 

927151 

133 

800277 

466 

199723 

44 

17 

727628 

333 

927071 

133 

800557 

466 

199443 

43 

18 

727828 

333 

926991 

133 

800836 

466 

199164 

42 

19 

728027 

333 

926911 

133 

801116 

466 

198884 

41 

20 

728227 

333 

926831 

133 

801396 

466 

198604 

40 

21 

9  .  728427 

332 

9.926751 

133 

9.801675 

466 

10.198325 

39 

22 

728626 

332 

926671 

133 

801955 

466 

198045 

38 

23 

728825 

332 

926591 

133 

802234 

465 

197766 

37 

24 

729024 

332 

926511 

134 

802513 

465 

197487 

36 

25 

729223 

331 

926431 

134 

802792 

465 

197208 

35 

26 

729422 

331 

926351 

134 

803072 

465 

196928 

34 

27 

729621 

331 

926270 

134 

803351 

465 

196649 

33 

28 

729820 

331 

926190 

134 

803630 

465 

196370  32 

29 

730018 

330 

926110 

134 

803908 

465 

196092 

31 

30 

730216 

330 

926029 

134 

804187 

465 

195813 

30 

31 

9.730415 

330 

9  .  925949 

134 

9.804466 

464 

10.195534 

29 

32 

730613 

330 

925868 

134 

804745 

464 

195255 

28 

33 

730811 

330 

925788 

134 

805023 

464 

194977 

27 

34 

731009 

329 

925707 

134 

805302 

464 

194698 

26 

35 

731206 

329 

925626 

134 

805580 

464 

194420 

25 

36 

731404 

329 

925545 

135 

805859 

464 

194141 

24 

37 

731602 

329 

925465 

135 

806137 

464 

193863 

23 

38 

731799 

329 

925384 

135 

806415 

463 

193585 

22 

39 

731996 

328 

925303 

135 

806693 

463 

193307 

21 

40 

732193 

328 

925222 

135 

806971 

463' 

193029 

20 

41 

9.732390 

328 

9.925141 

135 

9.807249 

463 

10.192751 

19 

42 

732587 

328 

925060 

135 

807527 

463 

192473 

18 

43 

732784 

328 

924979 

135 

807805 

463 

192195 

17 

44 

732980 

327 

924897 

135 

808083 

463 

191917 

16 

45 

733177 

327 

924816 

135 

808361 

463 

191639 

15 

46 

733373 

327 

924735 

136 

808638 

462 

191362 

14 

47 

733569 

327 

924654 

136 

808916 

462 

191084 

13 

48 

733765 

327 

924572 

136 

809193 

462 

190807 

12 

49 

733961 

326 

924491 

136 

809471 

462 

190529 

11 

50 

734157 

326 

924409 

136 

809748 

462 

190252 

10 

51 

9.734353 

326 

9.924328 

136 

9.810025 

462 

10.189975 

9 

52 

734549 

326 

924246 

136 

810302 

462 

189698!  8 

53 

734744 

325 

924164 

136 

810580 

462 

189420!  7 

54 

734939 

325 

924083 

136 

810857 

462 

189143 

6 

55 

735135 

325 

924001 

136 

81  J  134 

461 

188866 

5 

56 

735330 

325 

923919 

136 

811410 

461 

188590 

4 

57 

735525 

325 

923837 

136 

811687 

461 

188313 

3 

58 

735719 

324 

923755 

137 

811964 

461 

188036 

2 

59 

735914 

324 

923673 

137 

812241 

461 

187759 

1 

00 

73f>  !  09 

324 

923591 

137 

81S517 

461 

187483 

() 

Cosine 

|   Sine 

Cotang. 

Tang. 

M. 

57  Degrees. 


SINES  AND  TANGENTS.     (33  Degrees.) 


51 


M.I    Sine   |   D.   j   Cosine  |  D.  |   Tang. 

D. 

Cotanx.  | 

0 

9.7361091  324 

9.923591 

137 

9.812517 

461 

10.187482|60 

1 

736303   324 

923509 

137 

812794 

461 

187206 

59 

2 

736498 

324 

923427 

137 

813070 

461 

186930 

58 

3 

736692 

323 

923345 

137 

813347 

460 

186653 

57 

4 

736886 

323 

923263 

137 

813623 

460 

186377 

56 

ft 

737080 

323 

923181 

137 

813899 

460 

186101 

55 

•6 

737274 

323 

923098 

137 

814175 

460 

185825 

54 

7 

737467 

323 

923016 

137 

814452 

460 

185548 

53 

8 

737661 

322 

922933 

137 

814728 

460 

185272 

52 

9 

737855 

322 

922851 

137 

815004 

460 

184996 

51 

10 

738048 

322 

922768 

138 

815279 

460 

184721 

50 

11 

9.738241 

322 

9.922686 

138 

9.815555 

459 

10.184445 

49 

12 

738434 

322 

922603 

138 

815831 

459 

184169 

48 

13 

738627 

321 

922520 

138 

816107 

459 

183893 

47 

14 

738820 

321 

922438 

138 

816382 

459 

183618 

46 

15 

739013 

321 

922355 

138 

816658 

459 

183342 

45 

16 

739206 

321 

922272 

138 

816933 

459 

183067 

44 

17 

739393 

321 

922189 

138 

817209 

459 

182791 

43 

18 

739590 

320 

922106 

138 

817484 

459 

182516 

42 

19 

739783 

320 

922023 

138 

817759 

459 

182241 

41 

20 

739975 

320 

921940 

138 

818035 

458 

181965 

40 

21 

9.740167 

320 

9.921857 

139 

9.818310 

458 

10.181690 

39 

22 

740359 

320 

921774 

139 

818585 

458 

181415 

38 

23 

740550 

319 

921691 

139 

818860 

458 

181140 

37 

24 

740742 

319 

921607 

139 

819135 

458 

180865 

36 

25 

740934 

319 

921524 

139 

819410 

458 

180590 

35 

26- 

741125 

319 

921441 

139 

819684 

458 

180316 

34 

27 

741316 

319 

921357 

139 

819959 

458 

180041 

33 

28 

741508 

318 

921274 

139 

820234 

458 

179766 

32 

29 

741699 

318 

921190 

139 

820508 

457 

179492 

31 

30 

741889 

318 

921107 

139 

820783 

457 

179217 

30 

31 

9.742080 

318 

9.921023 

139 

9.821057 

457 

10.178943 

29 

32 

742271 

318 

920939 

140 

821332 

457 

178668 

28 

33 

742462 

317 

920856 

140 

821606 

457 

178394 

27 

34 

742652 

317 

920772 

140 

821880 

457 

178120 

26 

35 

742842 

317 

920688 

140 

822154 

457 

177846 

25 

36 

743033 

317 

920604 

140 

822429 

457 

177571 

24 

37 

743223 

317 

920520 

140 

82270S 

457 

177297 

23 

38 

743413 

316 

920436 

140 

822977 

456 

177023 

22 

39 

743602 

316 

920352 

140 

823250 

456 

176750 

21 

40 

743792 

316 

920268 

140 

823524 

456 

176476 

20 

41 

9.743982 

316 

9.920184 

140 

9.823798 

456 

10.176202 

19 

42 

744171 

316 

920099 

140 

824072 

456 

175928 

18 

43 

744361 

315 

920015 

140 

824345 

456 

175655 

17 

44 

744550 

315 

919931 

141 

824619 

456 

175381 

16 

45 

744739 

315 

919846 

141 

824893 

456 

175107 

15 

46 

744928 

315 

919762 

141 

825166 

456 

174834 

14 

47 

745117 

315 

919677 

141 

825439 

455 

174561 

13 

48 

745306 

314 

919593 

141 

825713 

455 

174287 

12 

49 

745494 

314 

919508 

141 

825986 

455 

174014 

11 

50 

745683 

314 

919424 

141 

826259 

455 

173741 

10 

51 

9.745871 

314 

9.919339 

141 

9.826532 

455 

1  Q.I  73468 

9 

52 

746059 

314 

919254 

141 

826805 

455 

173195 

8 

53 

746248 

313 

9191691  141 

827078 

455 

172922 

7 

54 

746436 

313 

919085 

141 

827351 

455 

172649 

6 

55 

746624 

313 

919000 

141 

827624 

455 

172376 

5 

56 

746812 

313 

918915 

142 

827897 

454 

172103 

4 

57 

746999 

313 

918830 

142 

828170 

454 

171830 

3 

58 

747187 

312 

918745 

142 

828442 

454 

171558 

2 

69 

747374 

312 

918659 

142 

828715 

454 

171285 

1 

60 

747562 

312 

918574 

142 

828987 

454 

171013 

0 

|  Cosine  [ 

Sine 

Cotang. 

Tang.   |  M. 

56  Degrees. 


(34  Degrees.")     A  TABLE  OF  LOGARITHMIC 


M.    Sine 

D.     Cosine 

D.  |   Tang.   \   D. 

Cotang.  j 

0 

9.747562 

312 

9.918574 

142 

9.828987 

454 

10.171013 

60 

1 

747749 

312 

918489 

142 

829260 

454 

170740 

59 

2 

747936 

312 

918404 

142 

829532 

454 

170468 

58 

3 

748123 

311 

918318 

142 

829805 

454 

170195 

57 

4 

748310 

311 

918233 

142 

830077 

454 

169923 

56 

5 

748497 

311 

918147 

142 

830349 

453 

169651 

55 

6 

748683 

311 

918062 

142 

830621 

453 

169379 

54 

7 

748870 

311 

917976 

143 

830893 

453 

169107 

53 

8 

749056 

310 

917891 

143 

831165 

453 

168835 

52 

9 

749243 

310 

917805 

143 

831437 

453 

168563 

51 

10 

749429 

310 

917719 

143 

831709 

453 

168291 

50 

11 

9.749615 

310 

9.917634 

143 

9.831981 

453 

10.168019 

49 

12 

749801 

310 

917548 

143 

832253 

453 

167747 

48 

13 

749987 

309 

917462 

143 

832525 

453 

167475 

47 

14 

750172 

309 

917376 

143 

832796 

453 

167^04 

46 

15 

750358 

309 

917290 

143 

833068 

452 

166932 

45 

16 

750543 

309. 

917204 

143 

833339 

452 

166661 

44 

17 

750729 

309 

917118 

144 

833611 

452 

166389 

43 

18 

750914 

308 

917032 

144 

833882 

452 

1661  IS 

42 

19 

751099 

308 

916946 

144 

834154 

452 

165846 

41 

20 

751284 

308 

916859 

144 

834425 

452 

165575 

40 

21 

9.751469 

308 

9.916773 

144 

9.834696 

452 

10.165304 

39 

22 

751654 

308 

916687 

144 

834967 

452 

165033 

38 

23 

751839 

308 

916600 

144 

835238 

452 

164762 

37 

24 

752023 

307 

916514 

144 

835509 

452 

164491 

36 

25 

752208 

307 

916427 

144 

835780 

451 

164220 

35 

26 

752392 

307 

916341 

144 

836051 

451 

163949 

34 

27 

752576 

307 

916254 

144 

836322 

451 

163678 

33 

28 

752760 

307 

916167 

145 

836593 

451 

163407 

32 

29 

752944 

306 

916081 

145 

836864 

451 

16313,6 

31 

30 

753128 

306 

915994 

145 

837134 

451 

162866 

30 

31 

9.753312 

306 

9.915907 

145 

9.837405 

451 

10.162595 

29 

32 

753495 

306 

915820 

145 

837675 

451 

162325 

28 

33 

753679 

306 

915733 

145 

837946 

451 

162054 

27 

34 

753862 

305 

915646 

145 

838216 

451 

161784 

26 

35 

754046 

305 

915559 

145 

838487 

450 

161513 

25 

36 

754229 

305 

915472 

145 

838757 

450 

161243 

24 

37 

754412 

305 

915385 

145 

839027 

450 

160973 

23 

38 

754595 

305 

915297 

145 

839297 

450 

160703 

22 

39 

754778 

304 

915210 

145 

839568 

450 

160432 

21 

40 

754960 

304 

915123 

146 

839838 

450 

160162 

20 

41 

9.755143 

304 

9.915035 

146 

9.840108 

450 

10.159892 

19 

42 

755326 

304 

914948 

146 

840378 

450 

159622 

18 

43 

755508 

304 

914860 

146 

840647 

450 

159353 

17 

44 

755690 

304 

914773 

146 

840917 

449 

159083 

16 

45 

755872 

303 

914685 

146 

841187 

449 

158813 

15 

46 

756054 

303 

914598 

146 

841457 

449 

158543 

14 

47 

756236 

303 

914510 

146 

841726 

449 

158274 

13 

48 

756418 

303 

914422 

146 

841996 

449 

158004 

12 

49 

756600 

303 

914334 

146 

842266 

449 

157734 

11 

50 

756782 

302 

914246 

147 

842535 

449 

157465 

10 

51 

9.756963 

302 

9.914158 

147 

9.842805 

449 

10.157195 

9 

52 

757144 

302 

914070 

147 

843074 

449 

156926 

8 

53 

757326 

302 

913982 

147 

843343 

449 

156657 

7 

54 

757507 

302 

913894 

147 

843612 

449 

156388 

6 

55 

757688 

301 

913806 

147 

843882 

448 

156118 

5 

56 

757869 

301 

913718 

147 

844151 

448 

155849 

4 

57 

758050 

301 

913630 

147 

844420 

448 

155580 

3 

58 

758230 

301 

913541 

147 

844689 

448 

155311 

2 

59 

758411 

301 

913453 

147 

844958 

448 

155042 

1 

60 

758591 

301 

913365 

147 

845227 

448 

154773 

0 

'T  Cosine 

Sine   j    |  Cotang.   |          Tang.   j  M. 

55  Degrees. 


SINES  AND  TANGENTS.     (35  Degrees.) 


53 


M.  |   Sine   |   D.     Cosine 

P.  |   Tane   |   D. 

Cotaiig.   | 

0 

9  .  75859  1 

301 

9.913365 

147 

9.845227 

448 

10.154773 

60 

1 

758772 

300 

913276 

147 

845496 

448 

154504 

59 

2 

758952 

300 

913187 

148 

845764 

448 

154236 

58 

3 

759132 

300 

913099 

148 

846033 

448 

153967 

57 

4 

759312 

300 

913010 

148 

846302 

448 

153698 

56 

5 

759492 

300 

912922 

148 

846570 

447 

153430 

55 

6 

759672 

299 

912833 

148 

846839 

447 

153161 

54 

7 

759852 

299 

912744 

148 

847107 

447 

152893 

53 

8 

760031 

299 

912655 

148 

847376 

447 

152624 

52 

9 

760211 

299 

912566 

148 

847644 

447 

152356 

51 

10 

760390 

299 

912477 

14S 

847913 

447 

152087 

5.0 

11 

9.760569 

298  19.912388 

148 

9.848181 

447 

10.151819 

49 

12 

760748 

298 

912299 

149 

848449 

447 

151551 

48 

13 

760927 

298 

912210 

149 

848717 

447 

151283 

47 

14 

761106 

298 

912121 

149 

848986 

447 

151014 

46 

15 

761285 

298 

912031 

149 

849254 

447 

150746 

45 

16 

761464 

298 

911942 

149 

849522 

447 

150478 

44 

17 

761642 

297 

911853 

149 

849790 

446 

150210 

43 

18 

761821 

297 

911763 

149 

850058 

446 

149942 

42 

19 

761999 

297 

911674 

149 

850325 

446 

149675 

41 

20 

762177 

297 

911584 

149 

850593 

446 

149407 

40 

21 

9.762356 

297 

9.911495 

149 

9.850861 

446 

10.149139 

39 

22 

762534 

296 

911405 

149 

851129 

446 

148871 

38 

23 

762712 

296 

911315 

150 

851396 

446 

148604 

37 

24 

762889 

296 

911226 

150 

851664 

446 

148336 

36 

25 

763067 

296 

911136 

150 

851931 

446 

148069 

35 

26 

763245 

296 

911046 

150 

852199 

446 

147801 

34 

27 

763422 

296 

910956 

150 

852466 

446 

147534 

33 

28 

763600 

295 

910866 

150 

852733 

445 

147267 

32 

29 

763777 

295 

910776 

150 

853001 

445 

146999 

31 

30 

763954 

295 

910686 

150 

853268 

445 

146732 

30 

31 

9.764131 

295 

9.910596 

150 

9.853535 

445 

10.146465 

29 

32 

764308 

295 

910506 

150 

853802 

445 

146198 

28 

33 

764485 

294 

910415 

150 

854069 

445 

145931 

27 

34 

764662 

294 

910325 

151 

854336 

445 

145664 

26 

35 

764838 

294 

910235 

151 

'854603 

445 

145397 

25 

36 

765015 

294 

910144 

151 

854870 

445 

145130 

24 

37 

765191 

294 

9100,54 

151 

855137 

445 

144863 

23 

38 

765367 

294 

909963 

151 

855404 

445 

144596 

22 

39 

765544 

293 

909873 

151 

855671 

444 

144329 

21 

40 

785720 

293 

909782 

151 

855938 

444 

144062 

20 

41 

9.765896 

293 

9.909691 

151 

9.856204 

444 

10.143796 

19 

42 

766072 

293 

909601 

151 

856471 

444 

143529 

18 

43 

766247 

293 

909510 

151 

856737 

444 

143263 

17 

44 

766423 

293 

909419 

151 

857004 

444 

142996 

16 

45 

766598 

292 

909328 

152 

857270 

444 

142730 

15 

46 

766774 

292 

909237 

152 

857537 

444 

142463 

14 

47 

766949 

292 

909146 

152 

857803 

444 

142197 

13 

48 

767124 

292 

909055 

152 

858069 

444 

141931 

12 

49 

767300 

292 

908964 

152 

858336 

444 

141664 

11 

50 

767475 

291 

908873 

152 

858602 

443 

141398 

10 

51 

9.767649 

291 

9.908781 

152 

9.858868 

443 

10.141132 

9 

52 

767824 

291 

908690 

152 

859134 

443 

140866 

8 

53 

767999 

291 

908599 

152 

859400 

443 

140600 

7 

54 

768173 

291 

908507 

152 

859666 

443 

140334 

6 

55 

768348 

290 

908416 

153 

859932 

443 

140068 

5 

56 

768522 

290 

908324 

153 

860198 

443 

139802 

4 

57 

768697 

290 

908233 

153 

860464 

443 

139536 

3 

58 

768871 

290 

908141 

153 

860730 

443 

139270 

2 

59 

769045 

290 

908049 

153 

860995 

443 

139005 

1 

60 

769219!  290 

907958 

153 

861261 

443 

138739 

0 

|   Cosine 

Sine   | 

Uotanj!.   | 

Tang. 

M. 

54  Degrees. 


54 


(36  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M. 

Sine 

D 

Cosine   |  1).  |   Tang.      D.   |  Colanp. 

0 

9.  7692  19 

290 

9.9079581  153 

9.861261 

443 

"10.138739 

60 

1 

769393 

289 

907866  153 

861527 

443 

138473 

59 

2 

769566 

289 

907774 

153 

861792 

442 

138208 

58 

3 

769740 

289 

907682 

153 

862058 

442 

137942 

57 

4 

769913 

289 

907590 

153 

862323 

442 

137677 

56 

5 

770087 

289 

907498 

153 

862589 

442 

137411 

55 

6 

770260 

288 

907406 

153 

862854 

442 

137146 

54 

7 

770433 

288 

907314 

154 

863119 

442 

136881 

53 

8 

770606 

288 

907222 

154 

863385 

442 

136615 

52 

9 

770779 

288 

907129 

154 

863650 

442 

136350 

51 

10 

770952 

288 

907037 

154 

863915 

442 

136085 

50 

11 

9.771125 

288 

9  906945 

154 

9.864180 

442 

10.135820 

49 

12 

771298 

287 

906852 

154 

864445 

442 

135555 

48 

13 

771470 

287 

906760 

154 

864710 

442 

135290 

47 

14 

771643 

287 

906667 

154 

864975 

441 

135025 

46 

15 

771815 

287 

906575 

154 

865240 

441 

134760 

45 

16 

771987 

287 

906482 

154 

865505 

441 

134495 

44 

17 

772159 

287 

906389 

155 

865770 

441 

134230 

43 

18 

772331 

286 

906296 

155 

866035 

441 

133965 

42 

19 

772503 

286 

906204 

155 

866300 

441 

133700 

41 

20 

772675 

286 

906111 

155 

866561 

441 

133436 

40 

21 

9.772847 

286 

9.906018 

155 

9.866829 

441 

10.133171 

39 

22 

773018 

286 

905925 

155 

867094 

441 

132906 

38 

23 

773190 

286 

905832 

155 

867358 

441 

132642 

37 

24 

773361 

285 

905739 

155 

867623 

441 

132377 

36 

25 

773533 

285 

905645 

155 

867887 

441 

132113 

35 

26 

773704 

285 

905552 

155 

868152 

440 

131848 

34 

27 

773875 

285 

905459 

155 

868416 

440 

131584 

33 

28 

774046 

285 

905366 

156 

868680 

440 

131320 

32 

29 

774217 

285 

905272 

15H 

868945 

440 

131055 

31 

30 

774388 

284 

905179 

156 

869209 

440 

130791 

30 

31 

9.774558 

284 

9.905085 

156 

9.869473 

440 

10.130527 

29 

32 

774729 

284 

904992 

156 

869737 

440 

130263 

28 

33 

774899 

284 

904898 

156 

870001 

440 

129999 

27 

34 

775070 

284 

904804 

156 

870265 

440 

129735 

26 

35 

775240 

284 

904711 

156 

870529 

440 

129471 

25 

36 

775410 

283 

904617 

156 

870793 

440 

129207 

24 

37 

775580 

283 

904523 

156 

871057 

440 

128943 

23 

38 

775750 

283 

904429 

157 

871321 

440 

128679 

22 

39 

775920 

283 

904335 

157 

871585 

440 

128415 

21 

40 

776090 

283 

904241 

157 

871849 

439 

128151 

20 

41 

9.776259 

283 

9.904147 

157 

9.872112 

439 

10.127888 

19 

42 

776429 

282 

904053 

157 

872376 

439 

127624 

18 

43 

776598 

282 

903959 

157 

872640 

439 

127360 

17 

44 

776768 

282 

903864 

157 

872903 

439 

127097 

16 

45 

776937 

282 

903770 

157 

873167 

439 

126833 

15 

46 

777106 

282 

903676 

157 

873430 

439 

1265YO 

14 

47 

777275 

281 

903581 

157 

873694 

439 

126306 

13 

48 

777444 

281 

903487 

157 

873957 

439 

126043 

12 

49 

777613 

281 

903392 

158 

874220 

439 

125780 

11 

50 

777781 

281 

903298 

1-58 

874484 

439 

'125516 

10 

51 

9.777950 

281 

9.903203 

158 

9.874747 

439 

10.125253 

9 

52 

778119 

281 

903108 

158 

875010 

439 

124990 

8 

53 

778287 

280 

903014 

158 

875273 

438 

124727 

7 

54 

778455 

280 

902919 

158 

875536 

438 

124464 

6 

55 

778624 

280 

902824 

158 

875800 

438 

124200 

5 

56 

778792 

280 

902729 

158 

876063 

438 

123937 

4 

57 

778960 

280 

902634 

158 

876326 

438 

123674 

3 

58 

779128 

280 

902539 

159 

876589 

438 

123411 

2 

59 

779295 

279 

902444 

159 

876851 

438 

123149 

1 

60 

779463 

279 

902349 

159 

877114 

438 

122888 

0 

.  Cosine 

I   Sine 

|   Cotang.  |       |    Tang.   J  M. 

53  Degrees. 


SINES    AND    TANGENTS.       (37  Degrees.) 


55 


M.     Sine      D.   |   Cosine    I).  |   T:IM<;      [).      Cotnnvr.  | 

0 

9  .  779463 

279 

9  .  902349 

159 

9.877114 

438 

10.  1228^6 

60 

1 

779631 

279 

902253 

159 

87737? 

438 

122623 

59 

9 

779798 

279 

902158 

159 

877640 

438 

122360 

58 

3 

779966 

279 

902063 

159 

877903 

438 

122097 

57 

4 

780133 

279 

901967 

159 

878165 

438 

121835 

56 

5 

780300 

278 

901872 

159 

878428 

438 

121572 

55 

6 

780467 

278 

901776 

159 

878691 

438 

121309 

54 

7 

780634 

278 

901681 

159 

878953 

437 

121047 

53 

8 

780801 

278 

901585 

159 

879216 

437 

120784 

52 

9 

780968 

278 

901490 

159 

879478 

437 

120522 

51 

10 

781134 

278 

901394 

160 

879741 

437 

120259 

50 

11 

9.781301 

277 

9.901298 

160 

9.880003 

437 

10.119997 

49 

12 

781468 

277 

901202 

160 

880265 

437 

119735 

48 

13 

781634 

277 

901106 

160 

880528 

487 

119472 

47 

14 

781800 

277 

901010 

160 

880790 

437 

119210 

46 

15 

781966 

277 

900914 

160 

881052 

437 

118948 

45 

16 

782132 

277 

900818 

160 

881314 

437 

118686 

44 

17 

782298 

276 

900722 

160 

881576 

437 

1  18424 

43 

18 

782464 

276 

900626 

160 

881839 

437 

118161 

42 

19 

782630 

276 

900529 

160 

882101 

437 

117899 

41 

20 

782796 

276 

900433 

161 

882363 

436 

117637 

40 

21 

9.782961 

276 

9.900337 

161 

9.882625 

436 

10.117375 

39 

22 

783127 

276 

900240 

161 

882887 

436 

117113 

38 

23 

783292 

275 

900144 

161 

883148 

436 

116852 

37 

24 

783458 

275 

900047 

161 

883410 

436 

116590 

36 

26 

783623 

275 

899951 

161 

883672 

436 

116328 

35 

25 

783788 

275 

899854 

161 

883934 

436 

116066 

34 

27 

783953 

275 

899757 

161 

884196 

436 

115804 

33 

28 

784118 

275 

899660 

161 

884457 

436 

115543 

32 

29 

784282 

274 

899564 

161 

884719 

436 

115281 

31 

30 

784447 

274 

899467 

162 

884980 

436 

115020 

30 

31 

9.784612 

274 

9.899370 

162 

9.885242 

436 

10.114758 

29 

32 

784776 

274 

899273 

162 

885503 

436 

114497 

28 

33 

784941 

274 

899176 

162 

885765 

436 

1  14235 

27 

34 

785105 

274 

899078 

162 

886026 

436 

113974 

26 

35 

785269 

273 

898981 

1&2 

886288 

436 

113712 

25 

36 

785433 

273 

898884 

162 

886549 

435 

113451 

24 

37 

785597 

273 

898787 

162 

886810 

435 

113190 

23 

38 

785761 

273 

898689 

162 

887072 

435 

112928 

22 

39 

785925 

273 

898592 

162 

887333 

435 

112667 

21 

40 

786089 

273 

898494 

163 

887594 

435 

112406 

20 

41 

9.786252 

272 

9.898397 

163 

9.887855 

435 

10.112145 

19 

42 

786416 

272 

898299 

163 

888116 

435 

111884 

18 

43 

786579 

272 

898202 

163 

888377 

435 

111623 

17 

44 

786742 

272 

898104 

1-63 

888639 

435 

111361 

16 

45 

786906 

272 

898006 

163 

888900 

435 

111100 

15 

46 

787069 

272 

897908 

163 

889160 

435 

110840 

14 

47 

787232 

271 

897810 

163 

889421 

435 

110579 

13 

48 

787395 

271 

897712 

163 

889682 

435 

110318 

12 

49 

787557 

271 

897614 

163 

889943 

435 

110057 

11 

50 

787720 

271 

897516 

163 

890204 

434 

109796 

10 

51 

9.787883 

271 

9.897418 

164 

9.890465 

434 

10.109535 

9 

52 

788045 

271 

897320 

164 

890725 

434 

109275 

8 

53 

788208 

271 

897222 

164 

890986 

434 

109014 

7 

54 

788370 

270 

897123 

164 

891247 

434 

108753 

6 

55 

788532 

270 

897025 

164 

891507 

434 

108493 

5 

56 

788694 

270 

896926 

164 

891768 

434 

,108232 

4 

57 

788856 

270 

896828 

164 

892028 

434 

107972 

3 

58 

789018 

270 

890729 

164 

892289 

434 

107711 

2 

59 

789180 

270 

896631 

164 

892549 

434 

107451 

1. 

60 

789342 

269 

896532 

164 

892810 

'434 

107190 

0 

Cosine   | 

Sine       |  Cotang.            Tang.   j 

52  Degrees 


56 


(38  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M. 

Sine 

D. 

Cosine    D. 

Tanj;. 

D. 

Cotang.  | 

0 

9.789342 

269 

9.89653:2 

164 

9.892810 

434 

10.  107190;  60 

1 

789504 

269 

896433 

165 

893070 

434 

106930 

59 

2 

789665 

269 

896335 

165 

893331 

434 

106669 

58 

3 

789827 

269 

896236 

165 

893591 

434 

106409 

57 

4 

789988 

269 

896137 

165 

893851 

434 

106149 

56 

5 

790149 

269 

896038 

165 

894111 

434 

105889 

55 

6 

790310 

268 

895939 

165 

894371 

434 

105629 

54 

7 

790471 

268 

895840 

165 

894632 

433 

105368 

53 

8 

790632 

268 

895741 

165 

894892 

433 

105108 

52 

9 

790793 

268 

895641 

165 

895152 

433 

104848 

51 

10 

.  790954 

268 

895542 

165 

895412 

433 

104588 

50 

11 

9.791115 

268 

9.895443 

166 

9.895672 

433 

10,104328 

49 

12 

791275 

267 

895343 

166 

895932 

433 

104068 

48 

13 

791436 

267 

895244 

166 

896192 

433 

103808 

47 

14 

791596 

267 

895145 

166 

896452 

433 

103548 

46 

15 

791757 

267 

895045 

166 

896712 

433 

103288 

45 

16 

791917 

267 

894945 

166 

896971 

433 

103029 

44 

17 

792077 

267 

894846 

166 

897231 

433 

102769 

43 

18 

792237 

266 

894746 

166 

897491 

433 

102509 

42 

19 

792397 

266 

894646 

166 

897751 

433 

102249 

41 

20 

792557 

266 

894546 

166 

898010 

433 

101990 

40 

21 

9.792716 

266 

9.894446 

167 

9.898270 

433 

10.101730 

39 

22 

792876 

266 

894346 

lt>7 

898530 

433 

101470 

38 

23 

793035 

266 

894246 

167 

898789 

433 

101211 

37 

24 

793195 

265 

894146 

167 

899049 

432 

100951 

36 

25 

793354 

265 

894046 

167 

899308 

432 

100692 

35 

26 

793514 

265 

893946 

167 

899568 

432 

100432 

34 

27 

793673 

265 

893846 

167 

899827 

432 

100173 

33 

28 

793832 

265 

893745 

167 

900086 

432 

099914 

32 

29 

793991 

265 

893645 

167 

900346 

432 

099654 

31 

30 

794150 

264 

893544 

167 

900605 

432 

099395 

30 

31 

9.794308 

264 

9.893444 

168 

9.900864 

432 

10.099136 

29 

32 

794467 

264 

893343 

168 

901124 

432 

098876 

28 

33 

794626 

264 

893243 

168 

901383 

432 

098617 

27 

34 

794784 

264 

893142 

168 

901642 

432 

098358 

26 

35 

794942 

264 

893041 

168 

901901 

432 

098099 

25 

36 

795101 

264 

892940 

168 

902160 

432 

097840 

24 

37 

795259 

263 

892839 

168 

902419 

432 

097581 

23 

38 

795417 

263 

892739 

168 

902679 

432 

097321 

22 

39 

795575 

263 

892638 

168 

902938 

432 

097062 

21 

40 

795733 

263 

892536 

168 

903197 

431 

096803 

20 

41 

9.795891 

263 

9.892435 

169 

9.903455 

431 

10.096545 

19 

42 

796049 

263 

892334 

169 

903714 

431 

096286 

18 

43 

796206 

263 

892233 

169 

903973 

431 

096027 

17 

44 

796364 

262 

892132 

169 

904232 

431 

095768 

16 

45 

796521 

262 

892030 

169 

904491 

431 

095509 

15 

46 

796679 

262 

891929 

169 

904750 

431 

095250 

14 

47 

796836 

262 

891827 

169 

905008 

431 

094992 

43 

48 

796993 

262 

891726 

169 

905267 

431 

094733 

12 

49 

797150 

261 

891624 

169 

905526 

431 

094474 

11 

50 

797307 

261 

891523 

170 

905784 

431 

094216 

10 

51 

9.797464 

261 

9.891421 

170 

9.906043 

431 

10.093957 

9 

52 

797621 

261 

891319 

170 

906302 

431 

093698 

8 

53 

797777 

261 

891217 

170 

906560 

431 

093440 

7 

54 

797934 

261 

891115 

170 

906819 

431 

093181 

6 

55 

798091 

261 

891013 

170 

907077 

431 

092923 

5 

56 

798247 

261 

890911 

170 

907336 

431 

092664 

4 

57 

798403 

260 

890809 

170 

907594 

431 

092406 

3 

58 

798560 

260 

890707 

170 

907852 

431 

092148 

2 

59 

798716 

260  . 

890605 

170 

908111 

430 

091889 

1 

60 

798872 

260 

890503 

170 

908369 

430 

091631 

0 

Cosine 

Sine 

Coiang. 

Tang.   1  M. 

51  Decrees. 


SINES  AND  TANGENTS.      (39  Degrees.) 


57 


M.  |    Sine 

n. 

C.isine    D.    Tana.  |   D.     Cotans. 

0 

9.798872   200 

9.890503 

170 

9.908369 

430 

10.091631 

60 

1 

7990281  260 

890400 

171 

908628 

430 

091372 

59 

2 

799184!  260 

890298 

171 

908886 

430 

091114 

58 

3 

799339;  259 

890195 

171 

909144 

430 

090856 

57 

4 

799495   259 

890093 

171 

909402 

430 

090598 

56 

5 

799651 

259 

889990 

171 

909660 

430 

090340 

55 

6 

799806 

259 

889888 

171 

909918 

430 

090082 

54 

7 

799962 

259 

88978,5 

171 

910177   430 

089823 

53 

8 

800117 

259 

889682 

171 

910435   430 

089565 

52 

9 

800272 

258 

.  889579 

171 

910693   430 

08930? 

51 

10 

800427 

258 

889477 

171 

910951   430 

089049 

50 

11 

9.800582 

258 

9.8893/4 

172 

9.911209;  430 

10.088791 

49 

12 

800737 

258 

889271 

172 

91U67 

430 

088533 

48 

13 

800892 

258 

889168 

172 

911724 

430 

088276 

47 

14 

801047 

258 

889064 

172 

911982   430 

088018 

46 

15 

801201 

258 

888961 

172 

912240 

430 

087760 

45 

16 

801356 

257 

8888£8 

172 

912498 

430 

087502 

44 

17 

801511 

257 

888755 

172 

912756 

430 

087244 

43 

18 

801665 

257 

888651 

172 

913014 

429 

0869S6 

42 

19 

801819 

257 

888548 

172 

913271 

429 

086729 

41 

20 

801973 

257 

888444 

173 

913529 

429 

086471 

40 

21 

9.802128 

257 

9.888341 

173 

9.913787 

429 

10.086213 

39 

22 

802282 

256 

888237 

173 

914044 

429 

085956 

38 

23 

802436 

256 

888134 

173 

914302 

429 

085698 

37 

24 

802589 

256 

888030 

173 

914560 

429 

085440 

36 

25 

802743 

256 

887926 

173 

914817 

429 

085183 

35 

26 

'  802897 

256 

887822 

173 

915075 

429 

084925 

34 

27 

803050 

256 

887718 

173 

915332 

429 

084668 

33 

28 

803204 

256 

887614 

173 

915590 

429 

084410 

32 

29 

803357 

255 

887510 

173 

915847 

429 

084153 

31 

30 

803511 

255 

887406 

174 

916104 

429 

083896 

30 

31 

9.803664 

255 

9.887302 

174 

9.916362 

429 

10,083638 

29 

32 

803817 

255 

887198 

174 

916619 

429 

083381 

28 

33 

803970 

255 

887093 

174 

9168771  429 

083123 

M 

34 

804123 

255 

886989 

174 

917134J  429 

082866 

26 

35 

804276 

254 

886885 

174 

917391   429 

082609 

25 

36. 

804428 

254 

886780 

174 

917648   429 

082352 

24 

<»>v 

804581 

254 

886676 

174 

917905   429 

082095 

23 

38 

804734 

254 

886571 

174 

918163   428 

081837 

22 

39 

804886 

254 

886466 

174 

918420!  428 

08^580 

21 

40 

805039 

254 

886362 

175 

918677   428 

081323 

20 

41 

9.805191 

254 

9.886257 

175 

9.918934 

428 

10.081066 

19 

42 

805343 

253 

886152 

175 

919191 

428 

080809 

18 

43 

805495 

253 

886047 

175 

919448 

428 

080552 

17 

44 

805647 

253 

885942 

175 

919705 

428 

080295 

16 

45 

805799 

253 

885837 

175 

919962 

428 

080038 

15 

46 

805951 

253 

885732 

175 

920219 

428 

079781 

14 

47 

806103 

253 

885627 

175 

920476 

428 

079624 

13 

48 

806254 

253 

885522 

175 

920733 

428 

079267 

12 

49 

806406 

252 

885416 

175 

920990 

428 

079010 

11 

50 

806557 

252 

885311 

176 

921247 

428 

078753 

10 

51 

9.806709 

252  9.885205 

176 

9.921503 

428 

10.078497 

9 

52 

806860 

252 

885100 

176 

921760!  428 

078240 

8 

53 

807011 

252 

884994 

176 

922017   428 

077983 

7 

54 

807163 

252 

884889 

176 

9222741  428 

077726 

6 

55 

807314 

S52 

884783 

176 

922530 

428 

077470 

5 

56 

807465 

251 

884677 

176 

922787 

428 

077213 

4 

57 

807615 

251 

884572 

176 

923044 

428 

076956 

3 

58 

807766 

251 

884466 

176 

923300i  428 

076700 

2 

59 

807917 

251 

884360 

176 

923557   427 

076443 

1 

60 

808067'  251 

884254 

177 

923813!  427 

076187 

'0 

Cosine  | 

Sine 

Cotansf.  | 

Tan?:.    M. 

50  Decrees. 

H 

58 


(40  Degrees.)     A,  TABLE  OP  LOGAKITHMIC 


M. 

S-He 

D. 

Cosine   |  D.  |   Tang. 

D. 

Cotang.   | 

0 

9.808067 

251 

9.884254 

177 

9.923813 

427 

10.076187 

60 

1 

808218 

251 

884148 

177 

924070 

427 

07593U 

59 

2 

808368 

251 

884042 

177 

924327 

427 

075673 

58 

3 

808519 

250 

883936 

177 

924583 

427 

075417 

57 

4 

808669 

250 

883829 

177 

924840 

427 

075160 

56 

5 

.  808819 

250 

883723 

177 

925096 

427 

074904 

55 

6 

808969 

250 

883617 

177 

925352 

427 

074648 

54 

7 

809119 

250 

883510 

177 

925609 

427 

07439  ] 

53 

8 

809269 

250 

883404 

177 

925865 

427 

074135 

52 

9 

809419 

249 

883297 

178 

926122 

427 

073878 

51 

10 

809569 

249 

883191 

178 

926378 

427 

073622 

50 

11 

9.809718 

249 

9.883084 

178 

9.926634 

427 

10.073366 

49 

12 

809868 

249 

882977 

178 

926890 

427 

073110 

48 

13 

810017 

249 

882871 

178 

927147 

427 

072853 

47 

14 

810167 

249 

882764 

178 

927403 

427 

072597 

46 

15 

810316 

248 

882657 

178 

927659 

427 

072341 

45 

16 

810465 

248 

882550 

178 

8E7915 

427 

072085 

44 

17 

.810614 

248 

882443 

178 

928171 

427 

071829 

43 

18 

810763 

248 

882336 

179 

928427 

427 

071573 

42 

19 

810912 

248 

882229 

179 

928683 

427 

071317 

41 

20 

811061 

248 

882121 

179 

928940 

427 

071060 

40 

21 

9.811210 

248 

9.882014 

179 

9.929196 

427 

10.070804 

39 

22 

811358 

247 

881907 

179 

929452 

427 

070548 

38 

23 

811507 

247 

881799 

179 

929708 

427 

070292 

37 

24 

811655 

247 

881692 

179 

929964 

426 

070036 

36 

25 

811804 

247 

881584 

179 

930220 

426 

069780 

35 

26 

811952 

247 

881477 

179 

930475 

426 

069525 

34 

27 

812100 

247 

881369 

179 

930731 

426 

069269 

33 

28 

812248 

247 

881261 

180 

930987 

426 

069013 

32 

29 

812396 

246 

881153 

180 

931243 

426 

068757 

31 

30 

812544 

246 

881046 

180 

931499 

426 

068501 

30 

3i 

9.812692 

246 

9.880938 

180 

9.931755 

426 

10.068245 

29 

32 

812840 

246 

880830 

180 

932010 

426 

067990 

•28 

33 

812988 

246 

880722 

180 

932266 

426 

067734 

27 

34 

813135 

246 

880613 

180 

932522 

426 

067478 

26 

35 

813283 

246 

880505 

180 

932778 

426 

067222 

25 

36 

813430 

245 

880397 

180 

933033 

426 

066967 

24 

37 

813578 

245 

880289 

181 

933289 

426 

066711 

23 

38 

813725 

245 

880180 

181 

933545 

426 

066455 

22 

39 

813872 

245 

880072 

181 

933800 

426 

066200 

21 

40 

814019 

245 

879963 

181 

934056 

426 

065944 

20 

41 

9.814166 

245 

9.879855 

181 

9.934311 

426 

10.065689 

19 

42 

814313 

245 

879746 

181 

934567 

426 

065433 

18 

43 

814460 

244 

879637 

181 

934823 

426 

065177 

17 

44 

814607!  244 

879529 

181 

935078 

426 

064922 

16 

45 

814753   244 

879420 

181 

935333 

426 

064667 

15 

46 

814900   244 

879311 

181 

935589 

426 

064411 

14 

47 

815046   244 

879202 

182 

935844 

426 

064156 

13 

48 

815193   244 

879093 

182 

936100 

426 

063900 

12 

49 

815339 

244 

878984 

182 

936355 

426 

063645 

11 

50 

815485 

243 

878875 

182 

936610 

426 

063390 

10 

51 

9.815631 

243 

9.878766 

182 

9.936866 

425 

10.063134 

•  & 

52 

815778 

243 

878656 

182 

937121 

425 

062879 

8 

53 

815924 

243 

878547 

182 

937376 

425 

062624 

7 

54 

816069 

243 

878438 

182 

937632 

425 

062368 

6 

55 

816215 

243 

878328 

182 

937887 

425 

062113 

5 

56 

8163G1 

243 

878219 

183 

938142 

425 

061858 

4 

57 

816507 

242 

878109 

183 

938398 

425 

061602 

3 

58 

816652 

242 

877999  183 

938653 

425 

061347 

2 

59 

816798 

242 

877890  183 

938908 

425 

061092 

1 

60 

8169431  242 

877780  183 

939163 

425 

060837 

0 

Cosinc- 

Sine   |    |  Cotang.            Tang.    M. 

49  Degrees. 


SINES  AND  TANGENTS.     (41  Degrees.)  N 


59 


M. 

Sine      D.     Cosine  |  D.    Tang.   |   D. 

Cotang. 

0 

9.816943 

242 

9.877780 

183|  9.939163 

425 

10.060837 

GO 

1 

817088 

242 

877670 

183 

939418 

425 

060582 

59 

2 

817233 

242 

877560 

183 

939673 

425 

060327 

58 

3 

817379 

242 

877450 

183 

•  939928 

425 

060072 

57 

4 

817524 

241 

877340 

183 

940183 

425 

059817 

56 

5 

817668 

241 

877230 

184 

940438 

425 

059562 

55 

6 

817813 

241 

877120 

184 

940694 

425 

059306 

54 

7 

817958 

241 

877010 

184 

940949 

425 

059051 

53 

8 

818103 

-241 

876899 

184 

941204 

425 

058796 

52 

9 

818247 

241 

876789 

184 

941458 

425 

058542 

51 

10 

818392 

241 

876678 

184 

941714 

425 

058286 

50 

11 

9.818536 

240 

9.876568 

184 

9.941968 

425 

10.058032 

49 

12 

818681 

240 

876457 

184 

942223 

425 

057777 

48 

13 

818825 

240 

876347 

184 

942478 

425 

057522 

47 

14 

818969 

240 

876236 

185 

942733 

425 

057267 

46 

15 

819113 

240 

876125 

185 

942988 

425 

057012 

45 

16 

819257 

240 

876014 

185 

943243 

425 

056757 

44 

17 

x8!940l 

240 

875904 

185 

943498 

425 

056502 

43 

18 

819545 

239 

875793 

185 

943752 

425 

056248 

42 

19 

819689 

239 

875682 

185 

944007 

425 

055993 

41 

20 

819832 

239 

875571 

185 

944262 

425 

055738 

40 

21 

9.819976 

239 

9.875459 

185 

9.944517 

^425 

10.055483 

39 

22 

820120 

239 

875348 

185 

944771 

424 

055229 

38 

23 

820263 

239 

875237 

185 

945026 

424 

054974 

37 

24 

820406 

239 

875126 

186 

945281 

424 

054719 

36 

25 

820550 

238 

875014 

186 

945535 

424 

054465 

35 

26 

820693 

238 

874903 

186 

945790 

424 

054210 

34 

27 

820836   238 

874791 

186 

946045 

424 

053955 

33 

28 

820*79 

238 

874680 

186 

946299 

424 

053701 

32 

29 

821122 

238 

874568 

186 

946554 

424 

053446 

31 

30 

821265 

238 

874456 

186 

946808 

424 

053192 

30 

31 

9.821407 

238 

9.874344 

186 

9.947063 

424 

10.052937 

29 

32 

821550 

238 

874232 

187 

947318 

424 

052682 

28 

33 

821693 

237 

874121 

187 

947572 

424 

052428 

27 

34 

821835 

237 

874009 

187 

947826 

424 

052174 

56 

35 

821977 

237 

873896 

187 

948081 

424 

051919 

25 

36 

822120 

237 

873784 

187 

948336 

424 

051664 

24 

37 

822262 

237 

873672 

187 

948590 

424 

051410 

23 

38 

822404 

237 

873560 

187 

948844 

424 

051156 

22 

39 

822546 

237 

873448 

187 

949099 

424 

050901 

21 

40 

822688 

236 

873335 

187 

949353 

424 

050647 

20 

41 

9.822*30 

236 

9.87322a 

187 

9.949607 

424 

10.050393 

19 

42 

822972 

236 

873110 

188 

949862 

424 

050  13S 

18 

43 

823114 

236 

872998 

188 

950116 

424 

049884 

17 

44 

823255 

236 

872885 

188 

950370 

424 

049630 

16 

45 

823397 

236 

872772 

188 

950625 

424 

049375 

15 

46 

823539 

236 

872659 

188 

950879 

424 

049121 

14 

47 

823680 

235 

872547 

188 

951133 

424 

048867 

16 

48 

823821 

235 

872434 

188 

951388 

424 

048612 

12 

49 

823963 

235 

872321 

188 

951642 

424 

048358 

11 

50 

824104 

235 

872208 

'188 

951896 

424 

048104 

10 

51 

9.824245 

235 

9.872095 

189 

9.952150 

424 

10.047850 

9 

52 

824386 

235 

871981 

189 

952405 

424 

047595 

8 

53 

824527 

235 

871§68 

189 

952659 

424 

047341 

7 

54 

824668 

234 

871755 

189 

952913 

424 

047087 

6 

55 

824808 

234 

871641 

189 

953107 

423 

046833 

5 

56 

824949 

234 

871528 

189 

953421 

423 

046579 

4 

57 

825090 

234 

871414 

189 

953675 

423 

046325 

3 

58 

825230 

?,34 

871301 

189 

953929 

423 

046071 

2 

59 

825371 

234 

871187 

189 

954183 

423 

045817 

1 

GO 

825511!  234 

87  i  073 

190 

954437 

423 

045563 

0 

Cosine  j          Sine        j  Colang   |          Tang. 

M. 

.  48  Degrees. 


60 


(42  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine      D.     Cosine    D.  |   Tanc. 

I).        ColHIlp.    | 

0 

9.825511 

234 

9.871073 

190 

9.954437 

423 

10.045563 

60 

1 

825651 

233 

870960 

190 

954691 

423 

045309 

<59 

2 

825791 

233 

870846 

J90 

954945 

423 

045055 

58 

3 

825931 

233 

870732 

J90 

955200 

423 

044800 

57 

4 

826071 

233 

870618 

190 

955454 

423 

044546 

56 

5 

826211 

233 

870504 

190 

955707 

423 

044293 

55 

6 

826351 

233 

870390 

190 

955961 

423 

044039 

54 

7 

826491 

233 

870276 

190 

956215 

423 

043785 

.53 

8 

826631 

233 

870161 

190 

956469 

423 

043531 

52 

9 

826770 

232 

87004? 

191 

956723 

423 

043277 

51 

10 

826910 

232 

869933 

191 

956977 

423 

043023 

50 

11 

9.827049 

232 

9.869818 

191 

9.957231 

423 

10.042769 

49 

12 

827189 

232 

869704 

191 

957485 

423 

042515 

48 

13 

827328 

232 

869589 

191 

957739 

423 

042261 

47 

14 

827467 

232 

869474 

191 

957993 

423 

042007 

46 

15 

827606 

232 

869360 

191 

958246 

423 

041754 

45 

16 

827745 

232 

869245 

191 

958500 

423 

041500 

44 

17 

827884 

231 

869130 

191 

958754 

423 

041246 

43 

18 

82&023 

231 

869015 

192 

959008 

423 

040992 

42 

19 

828162 

231 

868900 

192 

959262 

423 

040738 

41 

20 

828301 

231 

868785 

192 

959516 

423 

040484 

40 

21 

9.828439 

231 

9.868670 

192 

9.959769 

423 

10.040231 

39 

22 

828578 

231 

868555 

192 

960023 

423 

039977 

38 

23 

828716 

231 

868440 

192 

960277 

423 

039723 

37 

24 

828855 

230 

868324 

192 

960531 

423 

039469 

36 

25 

828993 

230 

868209 

192 

960784 

423 

039216 

35 

26 

829131 

230 

868093 

192 

961038 

423 

038962 

34 

27 

829269 

230 

867978 

193 

961291 

423 

038709 

33 

28 

829407 

230 

867862 

193 

961545 

423 

038455 

32 

29 

829545 

230 

867747 

193 

961799 

423 

038201 

31 

30 

829683 

230 

867631 

193 

962052 

423 

037948 

30 

31 

9.829821 

229 

9.867515 

193 

9.962306 

423 

10.037694 

29 

32 

829959 

229 

867399 

193 

962560 

423 

037440 

28 

33 

830097 

229 

867283 

193 

962813 

423 

037187 

27 

34 

830234 

229 

867167 

193 

963067 

423 

036933 

26 

35 

830372 

229 

867051 

193 

963320 

423 

036680 

25 

36 

830509 

229 

866935 

194 

963574 

423 

036426 

24 

37 

830646 

229 

866819 

194 

963827 

423 

036173 

23 

38 

830784 

229 

866703 

194 

964081 

423 

035919 

22 

39 

830921 

228 

866586 

194 

964335 

423 

035665 

21 

40 

831058 

228 

866470 

194 

964588 

422 

.  035412 

20 

41 

9.831195 

228 

9.866353 

194 

9  .  964842 

422 

10.035158 

19 

42 

831332 

228 

866237 

194 

965095 

422 

034905 

18 

43 

831469 

228 

866120 

194 

965349 

422 

034651 

17 

44 

831606 

228 

866004 

195 

965602 

422 

034:398 

16 

45 

831742 

228 

865887 

195 

965855 

422 

034145 

15 

46 

831879 

228 

865770 

195 

966109 

422 

033891 

14 

47 

832015 

227 

865653 

195 

966362 

422 

033638 

13 

48 

832152 

227 

865536 

195 

966616 

422 

033384 

12 

49 

832288 

227 

865419 

195 

966869 

422 

033131 

11 

50 

832425 

227 

865302 

195 

967123 

422 

032877 

10 

51 

9.832561 

227 

9.865185 

195 

9.967376 

422 

10.032624 

9 

52 

832697 

227 

865068 

195 

967629 

422 

032371 

8 

53 

832833 

227 

864950 

195 

967883 

422 

032117 

7 

54 

832969 

226 

864833 

196 

968136 

422 

031864 

6 

55 

833105 

226 

864716 

196 

968389 

422 

031611 

5 

56 

833241 

226 

864598 

196 

968643 

422 

031357 

4 

57 

833377 

226 

864481 

196 

968896 

422 

031104 

3 

58 

833512 

226 

864363 

196 

969149 

422 

030851 

2 

59 

833648 

226 

864245 

196 

969403 

422 

030597 

1 

60 

833783 

226 

864127 

196 

969656 

422 

030344 

0 

COM!  ne  ! 

Sine         Cotang.  | 

Tang. 

M. 

47  Degrees. 


SINES  AND  TANGENTS.      (43  Degrees.) 


61 


M 

Sine      1). 

Cosiiif;   |  D. 

T;ui».   j   D.      Cottinp.  j 

0 

9.833783 

226 

9.864127 

196 

9.969656 

422 

10.030344 

60 

1 

•  833919 

225 

864010 

196 

969909 

422 

030091 

59 

2 

834054 

225 

863892 

197 

970162 

422 

029838 

58 

3 

834189 

225 

863774 

197 

'  970416 

422 

029584 

57 

4 

834325 

225 

863656 

1-97 

970669 

422 

029331 

56 

5 

834460 

225 

863538 

197 

970922 

422 

029078 

55 

6 

834595 

225 

863419 

197 

971175 

422 

028825 

54 

7 

834730 

225 

863301 

197 

971429 

422 

028571 

53 

8 

834865 

225 

863183 

197 

971682 

422 

0283  18  1  52 

9 

834999 

224 

863064 

197 

971935 

422 

028065',  51 

10 

835134 

224 

862946 

198 

972188 

422 

027812  50 

11 

9.835269 

224 

9.862827 

198 

9.972441 

422 

10.027559 

49 

12 

835403 

224 

862709 

198 

972694 

422 

027306 

48 

13 

835538 

224 

862590 

198 

972948 

422 

027052 

47 

14 

835672 

224 

862471 

198 

973201 

422 

026799 

46 

15 

835807 

224 

862353 

198 

973454 

422 

026546 

45' 

16 

835941 

224 

862234 

198 

973707 

422 

026293 

44 

17 

836075 

223 

862115 

198 

973960 

422 

026040 

43 

18 

.  836209 

223 

861996 

198 

974213 

422 

025787 

42 

19 

836343 

223 

861877 

198 

974466 

422 

025534 

41 

20 

836477 

223 

861758 

199 

974719 

422 

025281 

40 

21 

9.836611 

223 

9.861638 

199 

9.974973 

422 

10.025027 

39 

22 

836745 

223 

861519 

199 

975226 

422 

024774 

38 

23 

836878 

223 

861400 

199 

975479 

422 

024521 

37 

24 

837012 

'222 

861280 

199 

975732 

422 

-  024268 

36 

25 

837146 

222 

861161 

199 

975985 

422 

024015 

35 

26 

837279 

222 

861041 

199 

976238 

422 

023762  34 

27 

837412 

222 

860922 

199 

976491 

422 

023509 

33 

28 

837546 

222 

860802 

199 

976744 

422 

023256 

32 

29 

837679 

222. 

860682 

200 

976997 

422 

023003 

31 

30 

837,81-2 

222 

860562 

200 

977250 

422 

022750 

30 

31 

9.837945 

222 

9.860442 

200 

9.977503 

422 

10.022497 

29 

32 

838078 

221 

860322 

200 

977756 

422 

022244 

28 

33 

838211 

221 

860202 

200 

978009 

422 

021991 

27 

34 

838344 

221 

860082 

200 

978262 

422 

021738 

26 

35 

838477 

221 

859962 

200 

978515 

422 

021485 

25 

36 

838610 

221 

859842 

200 

978768 

422 

021232 

24 

37 

838742 

221 

859721 

201 

979021 

422 

020979 

23 

38 

838875 

221 

859601 

201 

.  979274 

422 

020726 

22 

39 

839007 

221 

859480 

201 

979527 

422 

020473 

21 

40 

839140 

220 

859360 

201 

979780 

422 

020220 

20 

41 

9.839272 

220 

9.859239 

201 

9.980033 

422 

10.019967 

19 

42 

839404 

220 

859119 

201 

980286 

422 

019714 

18 

43 

839536 

220 

858998 

201 

980538 

422 

019462 

17 

44 

839668 

220 

858877 

201 

980791 

421 

019209 

16 

45 

839800 

220 

858756 

202 

981044 

421 

018956 

15 

46 

839932 

220 

858635 

202 

981297 

421 

018703 

14 

47 

840064 

219 

858514 

202 

981550 

421 

018450 

13 

48 

840196 

219 

858393 

202 

981803 

421 

018197 

12 

49 

840328 

219 

858272 

202 

982056 

421 

017944 

11 

50 

840459 

219 

858151 

202 

982309 

421 

017691 

10 

51 

9.840591 

219 

9.858029 

202 

9.982562 

421 

10.017438 

9 

52 

840722 

219 

857908 

202 

982814 

421 

017186 

8 

53 

840854 

219 

857786 

202 

983067 

421 

016933 

7 

54 

840985 

219 

857665 

203 

983320 

421 

016680 

6 

55 

841116 

218 

857543 

203 

983573 

421 

016427 

5 

56 

841247 

218 

857422 

203 

983826 

421 

016174 

4 

57 

841378 

218 

857300 

203 

984079 

421 

015921 

3 

58 

841509 

218 

857178 

203 

984331 

421 

015669 

2 

59 

841640 

218 

857056  203 

984584 

421 

015416|  1 

60 

841771   218 

856934|  203 

984837 

421 

015163'  0 

Cosine 

Sine   |    j   Cotang. 

'fang.   |  M. 

46  Degrees. 


(44  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine      1).     Cosine   |  1).     Tana.      D.     Cotang. 

0 

9.841771 

218 

9.856934 

203 

•9.984837 

421 

10.015163 

60 

1 

841902 

218 

856812 

203 

985090 

421 

0149  It) 

59 

2 

842033 

218 

856690 

204 

985343 

421 

014657 

58 

3 

842163 

'  217 

856568 

204 

985596 

421 

014404 

57 

4 

842294 

217 

856446 

204 

985848 

421 

014152 

56 

5 

842424 

217 

856323 

204 

986101 

421 

013899 

55 

6 

842555 

217 

858201 

204 

986354 

421 

013646 

54 

7 

842685 

217 

856078 

204 

986607 

421 

013393 

53 

8 

842815 

217 

855956 

204 

986860 

421 

013140 

52 

9 

842946 

217 

855833 

204 

987112 

421 

012888 

51 

10 

843076 

217 

855711 

205 

987365 

421 

012635 

50 

11 

9.843206 

216 

9.855588 

205 

9.987618 

421 

10.012382 

49 

12 

843336 

216 

855465 

205 

987871 

421 

012129 

48 

13 

843466 

216 

855342 

205 

988123 

421 

011877 

47 

14 

#43595 

216 

855219 

205 

988376 

421 

011624 

46 

15 

843725 

216 

855096 

205 

988629 

421 

011371 

45 

16 

843855 

216 

854973 

205 

988882 

421 

011118 

44 

17 

843984 

216 

854850 

205 

989134 

421 

010866 

43 

18 

844114 

215 

854727 

206 

989387 

421 

010613 

42 

19 

844243 

215 

854603 

206 

989640 

421 

010360 

41 

20 

844372 

215 

854480 

206 

989893 

421 

010107 

40 

21 

9.844502 

2l5 

9.854356 

206 

9.990145 

421 

10.009855 

39 

22 

844631 

215 

854233 

206 

990398 

421 

009602 

38 

23 

844760 

215 

854109 

206 

990651 

421 

009349 

37 

24 

844889 

215 

853986 

206 

990903 

421 

009097 

36 

25 

845018 

215 

853862 

206 

991156 

421 

008844 

35 

26 

845147 

215 

853738 

206 

991409 

421 

008591 

34 

.27 

845276* 

214 

853614 

207 

991662 

421 

008338 

33 

28 

845405 

214 

853490 

207 

991914 

421 

008086 

32 

29 

845533 

214 

853366 

207 

992167 

421 

007833 

31 

30 

845662 

214 

853242 

207 

992420 

421 

007580 

30 

31 

9.845790 

214 

9.853118 

207 

9.992672 

421 

10.007328 

29 

32 

845919 

214 

852994 

207 

992925 

421 

007075 

28 

33 

846047 

214 

852869 

207 

993178 

421 

00682'' 

27 

34 

846175 

214 

852745 

207 

993430 

421 

006570 

26 

35 

846304 

214 

852620 

207 

993683 

421 

006317 

25 

36 

846432 

213 

852496 

208 

993936 

421 

006064 

24 

37 

846560 

213 

852371 

208 

994189 

421 

005811 

23 

38 

846688 

213 

852247 

208 

994441 

421 

005559 

22 

39 

846816 

213 

852122 

208 

994694 

421 

005306 

21 

40 

846944 

213 

851997 

208 

994947 

421 

005053 

20 

41 

9.847,071 

213 

9.851872 

208 

9.995199 

421 

10.004801 

19 

42 

847199 

213 

851747 

208 

995452 

421 

004548 

18 

43 

847327 

213 

851622 

208 

995705 

421 

004295 

17 

44 

847454 

212 

851497 

209 

995957 

421 

004043 

16 

45 

847582 

212 

-  851372 

209 

996210 

421 

003790 

15 

46 

847709 

212 

851246 

209 

996463 

421 

003537 

14 

47 

847836 

212 

851121 

209 

996715 

421 

003285 

13 

48 

847964 

212 

850996 

209 

996968 

421 

003032 

12 

49 

848091 

212 

850870 

209 

997221 

421 

002779 

11 

50 

848218 

.  212 

850745 

209 

997473 

421 

002527 

10 

51 

9.848345 

212 

9.850619 

209 

9.997726  421 

10.002274 

9 

52 

848472 

211 

850493 

210 

997979 

421 

002021 

8 

53 

848599 

211 

850368 

210 

998231 

421 

001769 

7 

54 

848726 

211 

•  850242 

210 

998484 

421 

001516 

6 

5>5 

848852 

211 

850116 

210 

998737  421 

001263 

5 

56 

848979 

211 

849990 

210 

998989  421 

001011 

4 

57 

849106 

211 

849864 

210 

999242^  421 

000758 

3 

58 

843232 

211 

849738 

210 

999495 

421 

000505 

2 

59 

849359 

211 

849611 

210 

999748 

421 

000253 

1 

60 

849485   211 

849485 

210 

10.000000 

421 

000000  0 

Cosine  1 

Sine   | 

Co'itang.   1 

Tang.    |  M. 

45  Degrees. 


A  TRAVERSE    TABLE, 


SHOWING    THE   DIFFERENCE  OF 


LATITUDE   AND    DEPARTURE 


FOR  DISTANCES  BETWEEN    1  AND    100,  AND   FOB  ANGLES 
TO   Q^feTER   DEGREES   BETWEEN  1°  AND  90° 


TRAVERSE    TABLE. 


o 

£ 

4  Deg. 

t»» 

fDeg. 

ff 

1  § 

o 
p 

Dat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

1.00 

0.00 

1.00 

0.01 

1.00 

0.01 

I 

2 

2.00 

0.01 

2.00 

0.02 

2.00 

0.03 

2 

3 

3.00 

0.01 

3.00 

0.03 

3.00 

0.04 

3 

4 

4.00 

0.02 

4.00 

0.03 

4.00 

0.05 

4 

5 

5.00 

'0.02 

5.00 

0.04 

5.00 

0.07 

5 

6 

6.00 

0.03 

6.00 

0.05 

6.00 

0.08 

6 

7 

7.00 

0.03 

7.00 

0.06 

7.00 

0.09 

7 

8 

8.00 

0.03 

8.00 

0.07 

8.00 

0.10 

8 

9 

9.00 

0.04 

9.00 

0.08 

9.00 

0.12 

9 

10 

10.00 

0.04 

10.00 

0.09 

10.00 

.0.13 

10 

11 

11.00 

0.05 

11.00 

0.10 

11.00 

0.14 

ri 

12 

12.00 

0.05 

12.00 

0.10 

12.00 

0.16 

12 

13 

13.00 

0.06 

13.00 

0.11 

13.00 

0.17 

13 

14 

14.00 

0.06 

14.00 

0.12 

14.00 

0.18 

14 

15 

15.00 

0.07 

15.00 

0.13 

15.00 

0.20 

15 

16 

16.00 

0.07 

16.00 

0.14 

16.00 

0.21 

16 

17 

17.00 

0.07 

17.00 

0.15 

17.00 

0.22 

17 

18 

18.00 

0.08 

18.00 

0.16 

18.00 

0.24 

18 

19 

19.00 

0.08 

19.00 

0.17 

19.00 

0.25 

19 

20 

20.00 

0.09 

20.00 

0.17 

20.00 

0.26 

20 

21 

21.00 

0.09 

21.00 

0.18 

21.00 

0.27 

21 

22 

22,00 

0.10 

22.00 

0.19 

22.00 

0.29 

22 

23 

23.00 

0.10 

23.00 

0.20 

23.00 

0.30 

23 

24 

24.00 

0.10 

24.00 

0.21 

24.00 

0.31 

24 

25 

25.00 

0.11 

25.00 

0.22 

25.00 

0.33 

25 

26 

26.00 

0.11 

26.00 

0.23 

26.00 

0.34 

26 

27 

27.00 

0.12 

27.00 

0.24 

27.00 

0.35 

27 

28 

28.00 

0.12 

28.00 

0.24 

28.00 

0.37 

28 

29 

29.00 

0.13 

29.00 

0.25 

29.00 

0.38 

29 

30 

30.00 

0.13 

30.00 

0.26 

30.00 

0.39 

30 

31 

31.00 

0.14 

31.00 

0.27 

31.00 

0.41 

31 

32 

32.00 

0.14 

32.00 

0.28 

32.  0| 

0.42 

32 

33 

33.00 

0.14 

33.00 

0.29 

33.1" 

0.43 

33 

34 

34.00 

0.15 

34.00 

0.30 

34.00 

0.45 

34 

35 

35.00 

0.15 

35.00 

0.31 

35.00 

0.46 

35 

36 

36.00 

0.16 

36.00 

0.31 

36.00 

0.47 

36 

37 

37.00 

0.16 

37.00 

0.32 

37.00 

0.48 

37 

38 

38.00 

0.17 

38.00 

0.33 

38.00 

0.50 

38 

39 

39.00 

0.17 

39.00 

0.34 

39.00 

0.51 

39 

40 

40.00 

0.17 

40.00 

0.35 

40.00 

0.52 

40 

41 

41.00 

0.18 

41.00 

0.36 

41.00 

0.54 

41 

42 

42.00 

0.18 

42.00 

0.37 

42.00 

0.55 

42 

43 

43.00 

0.19 

43.00 

0.38 

43.00 

0.56 

43 

44 

44.00 

0.19 

44.00 

0.38 

44.00 

0.58 

44 

45 

45.00 

0.20 

45.00 

0.39 

45.00 

0.59 

45 

46 

46.00 

0.20 

46,00 

0.40 

46.00 

0.60 

46 

47 

47.00 

0.21 

47.00 

0.41 

47.00 

0.62 

47 

48 

48.00 

0.21 

48.00 

0.42 

48.00 

0.63 

48 

49 

49.00 

0.21 

49.00 

0.43 

49.00 

0.64 

49 

50 

50.00 

0.22 

50.00 

0.44 

50.00 

0.65 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

oJ 
o 
q 

a 

89|  Deg. 

89^  Deg. 

89i  Deg. 

S 

TRAVERSE    TABLE. 


g 

iDeg. 

iDeg. 

I  Deg. 

0 

B 

£ 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

51.00 

0.22 

51.00 

0.45 

51.00 

0.67~ 

51 

52 

52.00 

0.23 

52.00 

0.45 

52.00 

0.68 

52 

53 

53.00 

0.23 

53.00 

0.46 

53.00 

0.60 

53 

54 

54.00 

0.24 

54.00 

0.47 

54.00 

0.71 

54 

55 

55.00 

0.24 

55.00 

0.48 

55.00 

0.72 

55 

56 

56.00 

0.24 

56.00 

0.49 

56.00 

0.73 

56 

57 

57.00 

0.25 

57.00 

0.50 

57.00 

0.75 

57 

58 

58.00 

0.25 

58.00 

0.51 

57.99 

0.76 

58 

59 

59.00 

0.26 

59.00 

0.51 

58.99 

0.77 

59 

60 

60.00 

0.26 

60.00 

0.52 

59.99 

0.79 

60 

61 

61.00 

0.27 

61.00 

0.53 

60.99 

0.80 

*1 

62 

62.00 

0.27 

62.00 

0.54 

61.99 

0.81 

t>2 

63 

63.00 

0.27 

63.00 

0.55 

62.99 

0.82 

63 

64 

64.00 

0.28 

64.00 

0.56 

63.99 

0.84 

64 

65 

65.00 

0.28 

65.00 

0.57 

64.99 

0.85 

65 

66 

66.00 

0.29 

66.00 

0.58 

65.99 

0.86 

66 

67 

67.00 

0.29 

67.00 

0.58 

66.99 

0.88 

67 

63 

68.00 

0.30 

68.00 

0.59 

67.99 

0.89 

68 

69 

69.00 

0.30 

69.00 

0.60 

68.99 

0.90 

69 

70 

70.00 

0.31 

70.00 

0.61 

69.99 

0.92 

70 

71- 

71.00 

0.31 

71.00 

0.62 

70.99 

0.93 

71 

72 

72.00 

0.31 

72.00 

0.63 

71.99 

0.94 

72 

73 

73.00 

0.32 

73.00 

0.64 

72.99 

0.96 

73 

74 

74.00 

0.32  1 

74.00 

0.65 

73.99 

0.97 

74 

75 

75.00 

0.33 

75.00 

0.65 

74.99 

0,98 

75 

76 

76.00 

0.33 

76.00 

0.66 

75.99 

0.99 

76' 

77 

77.00 

0.34 

77.00 

0.67 

76.99 

.01 

77 

78 

78.00 

0.34 

78.00 

0.68 

77.99 

.02 

78 

79 

79.00 

0.34 

79.00 

0.69 

78.99 

.03 

79 

80 

80.00 

0.35 

80.00 

0.70 

79.99 

.05 

80 

81 

81.00 

0.35 

81.00 

0.71 

80.99 

.06 

81 

82 

82.00 

0.36 

82.00 

0.72 

81.99 

.07 

82 

83 

83.00 

0.36 

83.00 

0.72 

82.99 

.09 

83 

84 

84.00 

0.37 

84.00 

0.73 

83.99 

.10 

84 

85 

85.00 

0.37 

85.00 

0.74 

84.99 

.11 

85 

86 

86.00 

0.38 

86.00 

0.75 

85.99 

.13 

86 

87 

87.00 

0.38 

87.00 

0.76 

86.99 

.14 

87 

88 

88.00 

0.38 

88.00 

0.77 

87.99 

.15 

88 

89 

89.00 

0.39 

89.00 

0.78 

88.99 

.16 

89 

90 

90.00 

0.39 

90.00 

0.79 

89.99 

.18 

90 

91 

91.00 

0.40 

91.00 

0.79 

90.99 

.19 

91 

92 

92.00 

0.40 

92.00 

0.80 

91.99 

.20 

92 

93 

93.00 

0.41 

93.00 

0.81 

92.99 

.22 

93 

94 

94.00 

0.41 

94.00 

0.82 

93.99 

.23. 

94 

95 

95.00 

0.41 

95.00 

0.83 

94.99 

.24 

95 

96 

96.00 

0.42 

96.00 

0.84 

95.99 

.26 

96 

97 

97.00 

0.42 

97.00 

0.85 

96.99 

.27 

97 

98 

98.00 

0.43 

98.00 

0.86 

97.99 

.28 

98 

99 

99.00 

0.43 

99.00 

0.86 

98.99 

.30 

99 

100 

100.00 

0.44 

100.00 

0.87 

99.99 

.31 

100 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| 

.5 

I 

5 

89|  Deg. 

89£  Deg. 

89|  Deg. 

Q 

TRAVI.RSE    TABLE. 


o 

IDeg. 

H  Deg. 

11  Deg. 

H  Deg. 

5 

CQ 

1 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

1.00 

0.02 

1.00 

0.02 

1.00 

~0703~ 

1.00 

0.03 

i 

2 

2.00 

0.03 

2.00 

0.04 

2.00 

0.05 

2.00 

0.06 

2 

3 

3.00 

0.05 

3.00 

0.07 

3.00 

0.08 

3.00 

0.09 

3 

4 

4.00 

0.07 

4.00 

0.09 

4.00 

0.10 

4.00 

0.12 

4 

5 

5.00 

0.09 

5.00 

0.11 

5.00 

0.13 

5.00 

0.15 

5 

6 

6.00 

0.10 

6.00 

0.13 

6.00 

0.16 

6.00 

0.18 

6 

7 

7.00 

0.12 

7.00 

0.15 

7.00 

0.18 

7.00 

0.21 

7 

8 

8.00 

0.14 

8.00 

0.17 

8.00 

0.21 

8.00 

0.25 

8 

9 

9.00 

0.16 

9.00 

0.20 

9.00 

0.24 

9.00 

0.28 

9 

10 

10.00 

0.17 

10.00 

0.22 

10.00 

0.26 

10.00 

0.31 

10 

11 

11.00 

0.19 

11.00 

0.24 

11.00 

0.28 

10.99 

0.34 

11 

12 

12.00 

0.21 

12.00 

0.26 

12.00 

0.31 

11.99 

0.37 

12 

13 

13.00 

0.23 

13.00 

0.28 

13.00 

0.34 

12.99 

0.40 

13 

14 

14.00 

0.24 

14.00 

0.31 

14.00 

0.37 

13.99 

0.43 

14 

15 

15.00 

0.26 

15.00 

0.33 

14.99 

0.39 

14.99 

0.46 

15 

16 

16.00 

0.28 

16.00 

0.35 

15.99 

0.42 

15.99 

0.49 

16 

17 

17.00 

0.30 

17.00 

0.37 

16.99 

0.45 

16.99 

0.52 

17 

18 

18.00 

0.31 

18.00 

0.39 

17.99 

0.47 

17.99 

0.55 

18 

19 

19.00 

^0.33 

19.00 

0.41 

18.99 

0.50 

18.99 

0.58 

19 

20 

20.00 

0.35 

20.00 

0.44 

19.99 

0.52 

19.99 

0.61 

20 

21 

21.00 

0.37 

21.00 

0.46 

20.99 

0.55  j 

20.99 

0.64 

21 

22 

22.00 

0.38 

21.99 

0.48 

21.99 

0.58 

21.99 

0.67 

22 

23 

23.00 

0.40 

22.99 

0.50 

22.99 

0.60 

22.99 

0.70 

23 

24 

24.00 

0.42 

23.99 

0.52 

23.99 

0.63 

23.99 

0.73 

24 

25 

25.00 

0.44 

24.99 

0.55 

24,99 

0.65 

24.99 

0.76 

25 

26 

26.00 

0.45 

25.99 

0.57 

25.99 

0.68 

25.99 

0.79 

26 

27 

27.00 

0.47 

26.99 

0  59 

26.99 

0.71 

26.99 

0.83 

27 

28 

28.00 

0.49 

27.99 

0.61 

27.99 

0.73 

27.99 

0.86 

28 

29 

29.00 

0.51 

28.99 

0.63 

28.99 

0.76 

28*.  99 

0.89 

29 

30 

30.00 

0.52 

29.99 

0.65 

29.99 

0.79 

29.99 

0.92 

30 

31 

31.00 

0.54 

30.99 

0.68 

30.99 

0.81 

30.99 

0.95 

31 

32 

32.00 

0.56 

31.99 

0.70 

31.99 

0.84 

31.99 

0.98 

32 

33 

32.99 

0.58 

32.99 

0.72 

32  .  99 

0.86 

32.98 

1.01 

33 

34 

33.99 

0.59 

33.99 

0.74 

33.99 

0.89 

33.98 

1.04 

34 

35 

34.99 

0.61 

34.99 

0.76 

34.99 

0.92 

34.98 

1.07 

35 

36 

35.99 

0.63 

35.99 

0.79 

35.99 

0.94 

35.98 

1.10 

36 

37 

36.99 

0.65 

36.99 

Q.81 

36.99 

0.97 

36.98 

1.13 

37 

38 

37.99 

0.66 

37.99 

0.83 

37.99 

0.99 

37.98 

1.16 

38 

39 

38.99 

0.68 

38.99 

0.85 

38.99 

1.02 

38.98 

1.19 

39 

40 

39.99 

0.70 

39.99 

0.87 

39  .  99 

1.05 

39.98 

1.22 

40 

41 

40.99 

0.72 

40.99 

0.89 

40.99 

1.07 

40.98 

1.25 

'41 

42 

41.99 

0.73 

41.99 

0.92 

41.99 

1.10 

41.98 

1.28 

42 

43 

42.99 

0.75 

42.99 

0.94 

42.99 

1.13 

42.98 

1.31 

43 

44 

43.^9 

0.77 

43.99 

0.96 

43.99 

1.15 

43.98 

1.34 

44 

45 

44.99 

0.79 

44.99 

0.98 

44.99 

1.18 

44.98 

1.37 

45 

46 

45.99 

0.80 

45.99 

1.00 

45.99 

1.20 

45.98 

1.40 

46 

47 

46  .  99 

0.82 

46.99 

1.03 

46.99 

1.23 

46.98 

1.44 

47 

48 

47.99 

0.84 

47.99 

1.05 

47.98 

1.26 

47.98 

1.47 

48 

49 

48.99 

0.86 

48.99 

1.07 

48.98 

1.28 

48.98 

1.50 

49 

50 

49.99 

0.87 

49.99 

1.09 

49.98 

1.31 

49.98 

1.53 

50 

0> 
1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

.2 

b 

89  Deg. 

88|  Deg. 

881  De%. 

884  Deg. 

2 

TRAVERSE    TABLE. 


6 

P* 

IDeg. 

U  Deg. 

H  Des- 

HDeg. 

O 
1 

P 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

50.99 

0.89 

50.99 

.11 

50.98 

1.34 

50.98 

1.56 

51 

52 

51.99 

0.91 

51.99 

.13 

51.98 

.36 

51.98 

1.59 

52 

53 

52.99 

0.92 

52.99 

.10 

52.98 

.39 

52.98 

1.62 

53 

54 

53.99 

0.94 

53.99 

.18 

53.98 

.41 

53.97 

1.65 

54 

55 

54.99 

0.96 

54.99 

.20 

54.98 

.44 

54.97 

1.68 

55 

56 

55.99 

0.98 

55.99 

.22 

55.98 

.47 

55.97 

1.71 

56 

57 

56.99 

0.99 

56.99 

.24 

56.98 

.49 

56.97 

1.74 

57 

58 

57.99 

l.Ql 

57.99 

1.27 

57.98 

.52 

57.97 

1.77 

58 

59 

58.99 

1.03 

58.99 

1.29 

58.98 

.54 

58.97 

1.80 

59 

60 

59.99 

1.05 

59.99 

1.31 

59.98 

.57 

59.97 

1.83 

60 

61 

60.99 

1.06 

60.99 

.33 

60.98 

.60 

60.97 

1.86 

61 

62 

61.99 

1.08 

61.99 

.35 

61.98 

.62 

61.97 

1.89 

62 

63 

62.99 

1.10 

62.99 

.37 

62.98 

.65 

62.97 

1.92 

63 

64 

63.99 

.12 

63.98 

.40 

63.98 

.68 

63.97 

1.95 

64 

65 

64.99 

.13 

64.98 

.42 

64.98 

.70 

64.97 

1.99 

65 

66 

65.99 

.15 

65.98 

.44 

35.98 

.73 

65.97 

2.02 

66 

67 

66.99 

.17 

66.98 

.46 

66.98 

.75 

66.97 

2.05 

67 

68 

67.99 

.19 

67.98 

.48 

67.98 

.78 

67.97 

2.08 

68 

69 

68.99 

.20 

68.98 

1.51 

68.98 

.81 

68.97 

2.11 

69 

70 

69.99 

.22 

69.98 

1.53 

69.98 

.83 

69.97 

2.14 

70 

71 

70.99 

.24 

70.  9S 

1.55 

70.98 

.86 

70.97 

2.17 

71 

72 

71.99 

.26 

71.98 

1.57 

71.98 

.88 

71.97 

2.20 

72 

73 

72.99 

.27 

72.98 

1.59 

72.97 

.91 

'"2.97 

2.23 

73 

74 

73.99 

.29 

73.98 

1.61 

73.97 

.94! 

73.97 

2.26 

74 

75 

74.99 

.31 

74.98 

1.64 

74.97 

.96 

74.97 

2.29 

75 

76 

75.99 

.33 

75.98 

1.66 

75.97 

.99 

75.96 

2.32 

76 

77 

76.99 

.34 

76.98 

1.68 

76.97 

2.02 

76.96 

2.35 

77 

78 

77.99 

.36 

77.98 

1.70 

77.97 

2.04 

77.96 

2.38 

78 

79 

78.99 

.38 

78.98 

1.72 

78.97 

2.07 

78.96 

2.41 

79 

80 

79.99 

.40 

79.98 

1.75 

79.97 

2,09 

79.96 

2.44 

80 

81 

80.99 

.41 

80.98 

1.77 

80.97 

2.12 

80.96 

2.47 

81 

82  81.99 

.43 

81.98 

1.79 

81.97 

2.15 

81.96 

2.50 

82 

83 

82.99 

.45 

82.98 

1.81 

82.97 

2.17 

82.96 

2.53 

83 

84 

83.99 

.47 

83.98 

1.83 

83.97 

2.20 

83.96 

2.57 

84 

85 

84.99 

.48 

84.98 

1.85 

84.97 

2.23 

84.96 

2.60 

85 

86 

85.99 

.50 

85.98 

1.88 

85.97 

2.25 

85.96 

2.63 

86 

87 

86.99 

.52 

86.98 

1.90 

86.97 

2.28 

86.96 

2.66 

87 

88 

87.99 

.54 

87.98 

1.92 

87.97 

2.30 

87.96 

2.69 

88 

89 

88.99 

.55 

88.98 

1.94 

88.97 

2.33 

88.96 

2.72 

89 

90 

89.99 

.57 

89.98 

1.96 

89.97 

2.36 

89.96 

2.75 

90 

91 

90.99 

.59 

90.98 

1.99 

90.97. 

?.38 

90.96 

2.78 

91 

92 

91.99 

.61 

91.98 

2.01 

91.97 

2.41 

91.96 

2.81 

92 

93 

92.99 

.62 

92.98 

2.03 

92.97 

2.43 

92.96 

2.84 

93 

94 

93.99 

.64 

93.98 

2.05 

93.97 

2.46 

93.96 

2.87 

94 

95 

94.99 

.66 

94.98 

2.07 

94.97 

2.49 

94.96 

2.90 

95 

96 

95.99 

.68 

95.98 

2.09 

95.97 

2.51 

95.96 

2.94 

96 

97 

96.99 

.69 

96.98 

2.12 

96.97 

2*54 

96.95 

2.96 

97 

98 

97.99 

.71 

97.98 

2.14 

97.97 

2.57 

97-.  95 

2.99 

98 

99 

98.98 

.73 

98.98 

2.16 

98.97 

2.59 

98.95 

3.02 

99 

100 

99.98 

.75 

99.98 

2.18 

99.97 

2.62 

99.95 

3.05 

100 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

5 

89  Deg. 

88J  Deg. 

88£  Deg. 

88}  Deg. 

Q 

TRAVERSE    TABLE. 


•1 

2  Deg. 

2k  Deg. 

2£  Deg. 

2|  Deg. 

0 
s; 

P 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

1.00 

0.03 

1.00 

0.04 

1.00 

0.04 

1.00 

0.05 

l 

2 

2.00 

0.07 

2.00 

0.08 

2.00 

0.09 

2.00 

0.10 

2 

p 

3.00 

0;10 

3.00 

0.12 

3.00 

0.13 

3.00 

0.14 

3 

4 

4.00 

0.14 

4.00 

0.16 

4.00 

0.17 

4.00 

0.19 

4 

5 

5.00 

0.17 

5.00 

0.20 

5.00 

0.22 

4.99 

0.24 

5 

6 

6.00 

0.21 

6.00 

0.24 

5.99 

0.26 

5.99 

0.29 

6 

7 

7.00 

0.24 

6.99 

0.27 

6.99 

0.31 

6.99 

0.34 

7 

8 

7.99 

0.28 

7.99 

0.31 

7.99 

0.35 

7.99 

0.38 

8 

9 

8.99 

0.31 

8.99 

0.35 

8.99 

0.39 

8.99 

0.43 

9 

10 

9.99 

0.35 

9.99 

0.39 

9.99 

0.44 

9.99 

0.48 

10 

11 

10.99 

0.38 

10.99 

0.43 

10.99 

0.48 

10.99 

0.53 

11 

12 

11.99 

0.42 

11.99 

0.47 

11.99 

0.52 

11.99 

0.58 

12 

13 

12.99 

0.45 

12.99 

0.51 

12.99 

0.57 

12.99 

0.62 

13 

14 

13.99 

0.49 

13.99 

0.55 

13.99 

0.61 

13.98 

0.67 

14 

15 

14.99 

0.52 

14.99 

0.59 

14.99 

0.65 

14.98 

0.72 

15 

16 

15.99 

0.56 

15.99 

0.63 

15.99 

0.70 

15.98 

0.77' 

16 

17 

16.99 

0.59 

16.99 

0.67 

16.98 

0.74 

16.98 

0.82 

17 

18 

17.99 

0.63 

17.99 

0.71 

17.98 

0.79 

17.98 

0.86 

18 

19 

18.99 

0.66 

18.99 

0.75 

18.98 

0.83 

18.98 

0.91 

19 

20 

19.99 

0.70 

19.9? 

0.79 

19.98 

0.87 

19.98 

0.96 

20 

21 

20.99 

0.73 

20.98 

0.82 

20.98 

0.92 

20.98 

.01 

21 

22 

21.99 

0.77 

21.98 

0.86 

21.98 

0.96 

21.97 

.06 

22 

23 

22.99 

O.bO 

22.98 

0.90 

22.98 

1.00 

22.97 

.10 

23 

24 

23.99 

0.84 

23.98 

0.94 

23.98 

1.05 

23.97 

.15 

24 

25 

24.98 

0.87 

24.98 

0.98 

24.98 

1.09 

24.97 

.20 

25 

26 

25.98 

0.91 

25.98 

1.02 

25.98 

1.13 

25.97 

.25 

26 

27 

26.98 

0.94 

26.98 

1.06 

26.97 

1.18 

26.97 

.30 

27 

28 

27.98 

0.98 

27.98 

.10 

27.97 

1.22 

27.97 

.34 

28 

29 

28.98 

1.01 

28.98 

.14 

28.97 

1.26 

28.97 

.39 

29 

30 

29.98 

1.05 

29.98 

.18 

29  .  97 

1.31 

29  .  97 

.44 

30 

31 

30.98 

.08 

30.98 

.22 

30.9-7 

1.35 

30.96 

.49 

31 

32 

31.98 

.12 

31.98 

.26 

31.97 

1.40 

31.96 

.54 

32 

33 

32.98 

.15 

32.97 

.30 

32.97 

1.44 

32.96 

.58 

33 

34 

33.98 

.19 

33.97 

.33 

33.97 

1.48 

33.96 

.63 

34 

35 

34.98 

.22 

34.97 

.37 

34.97 

1  .  53 

34.96 

.68 

35 

36 

35.98 

.26 

35.97 

.41 

35.97 

1.57 

35  .  96 

.73 

36 

37 

36.98 

.29 

36.97 

.45 

36.96 

1.61 

36.96 

.78 

37 

38 

37.98 

.33 

37.97 

.49 

37.96 

1.66 

37.96 

.82 

38 

39 

38.98 

.36 

38.97 

.53 

38.96 

1.70 

38.96 

.87 

39 

40 

39.98 

.40 

39.97 

.57 

39.96 

1.75 

39  .  95 

1.92 

40 

41 

40.98 

.43 

40.97 

.61 

40.96 

1.77 

40.95 

1.97 

41 

42 

41.97 

.47 

41.97 

.65 

41.96 

1.83 

41.95 

2.02 

42 

43 

42.97 

.50 

42.97 

.69 

42.96 

1.88 

42.95 

2.06 

43 

44 

43.97 

.54 

43.97 

.73 

43.96 

1.92 

43.95 

2.11 

44 

45 

44.97 

.57 

44.97 

.77 

44.96 

1.96 

44.95 

2.16 

45 

46 

45.97 

.61 

45.96 

.81 

45.96 

2.01 

45.95 

2.21 

46 

47 

46.97 

.64 

46.96 

.85 

46.96 

2.05 

46.95 

2.25 

47 

48 

47.97 

.68 

47.96 

.88 

47.95 

2.09 

47.95 

2.30 

48 

49 

48.97 

1.71 

48.96 

.92 

48.95 

2.14 

48.94 

2.35 

49 

50 

49.97 

1.74 

49.96 

.96 

49.95 

2.18 

49.94 

2.40 

50 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

1 

88  Peg. 

87|  Deg. 

»  87^  Deg. 

87$  Deg. 

'to 

3 

TRAVERSE   TABLE. 


Q 

2  Deg. 

2*  Deg. 

2*  Deg. 

2J  Deg. 

G 

stance. 

f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

50.97 

1.78 

50.96 

2.00 

50.95 

2.22 

50.94 

2.45 

51 

52 

51.97 

1.81 

51.96 

2.04 

51.95 

2.27 

51.94 

2.50 

52 

53 

52.97 

1.85 

52.96 

2.08 

52.95 

2.31 

52.94 

2.54 

53 

54 

53.97 

1.88 

53.96 

2.12 

53.95 

2.36 

53.94 

2.59 

54 

55 

54.97 

1.92 

54.96 

2.16 

54.95 

2.40 

54.94 

2.64 

55 

56 

55.97 

1.95 

55.96 

2.  "20 

55.95 

2.44 

55.94 

2.69 

56 

57 

56.97 

1.99 

56.96 

2.24 

56.95 

2.49 

56.93 

2.73 

57 

58 

57.96 

2.02 

57.96 

2.28 

57.94 

2.53 

57.93 

2.78 

58 

59 

58.96 

2.06 

58.95 

2.32 

58.94 

2.57 

58.93 

2.83 

59 

60 

59.96 

2.09 

59.95 

2.36 

59.94 

2.62 

59.93 

2.88 

60 

61 

60.96 

2.13 

60,95 

2.39 

~60^4~ 

2.66 

60.93 

2.93 

61 

62 

61.96 

2.16 

61.95 

2.43 

61.94 

2.70 

61.93 

2.97 

62 

63 

62.96 

2.20 

62.95 

2.47 

62.94 

2.75 

62.93 

3.02 

63 

64 

63.96. 

2.23 

63.95 

2.51 

63.94 

2.79 

63.93 

3.07 

64 

65 

64.  96~ 

2.27 

64.95 

2.55 

64.94 

2;  84 

64.93 

3.12 

65 

66 

65.96 

2.30 

65.95 

2.59 

65.94 

2.88 

65.92 

3.17 

66 

67 

66.96 

2.34 

66.95 

2.63 

66.94 

2.92 

66.92 

3.21 

67 

68 

67.96 

2.37 

67.95 

2.67 

67.94 

2.97 

67.92 

3.26 

68 

69 

68.96 

2.41 

68.95 

2.71 

68.93 

3.01 

68.92 

3.31 

69 

70 

69.96 

2.44 

69.95 

2.75 

69.93 

3.05 

69.92 

3.36 

70 

71 

70.96 

2.481 

70.95 

2.79 

70.93 

3.10 

70.92 

3.41 

71 

72 

71.96 

2.51 

71.94 

2.83 

71.93 

3.14 

71.92 

3.45 

72 

73 

72.96 

2.55 

72.94 

2.87 

72.93 

3.18 

72.92 

3.50 

73 

74 

73.95 

2.58 

73.94 

2.91 

73.93 

3.23 

73.91 

3.55 

74 

75 

74.95 

2.62 

74.94 

2.94 

74.93 

3.27 

74.91 

3.60 

75 

76 

75.95 

2.65 

75.94 

2.98 

75.93 

3.31 

75.91 

3.65 

76 

77 

76.95 

2.69 

76.94 

3.02 

76.93 

3.36 

76.91 

3.70 

77 

78 

77.95 

2.72 

77.94 

3.06 

77.93 

3.40 

77.91 

3.74 

78 

79 

78.95 

2.76 

78.94 

3.10 

78.92 

3.45 

78.91 

3.79 

79 

80 

79.95 

2.79 

79.94 

3.14 

79.92 

3.49 

79.91 

3.84 

80 

81 

80.95 

2.83 

80.94 

3.18 

80.92 

3.53 

80.91 

3.89 

81 

82 

81.95 

2.86 

81/94 

3.22 

81.92 

3.58 

81.91 

3.93 

82 

83 

82.95 

2.90 

82.9>i 

3.26 

82.92 

3.62 

82.90 

3.98 

83 

84 

83.95 

2.93 

83.94 

3.30 

83.92 

3.66 

83:90 

4.03 

84 

85 

84.95 

2.97 

84.93 

3.34 

84.92 

3.71 

84.90 

4.08 

85 

86 

85.95 

3.00 

85.93 

3.38 

85.92 

3.75 

85.90 

4.13 

86 

87 

86.95 

3.04 

86.93 

3.42 

86.92 

3.79 

86.90 

4.17 

87 

88 

87.95 

3.07 

87.93 

3.45 

87.92 

3.84 

87.90 

4.22 

88 

89 

88.95 

3.11 

88.93 

3.49 

88.92 

3.88 

88.90 

4.27 

89 

90 

89.95 

3.14 

89.93 

3.53 

89.91 

3.93 

89.90 

4.32 

90 

91 

90.95 

3:18 

90.93 

3.57 

90.91 

3.97 

90.90 

4.37 

91 

92 

91.94 

3.21 

91.93 

3.61 

91.91 

4.01 

91.89 

4.41 

92 

93 

92.94 

3.25 

92.93 

3.65 

92.91 

4.06 

92.89 

4.46 

93 

94 

93.94 

3.28 

93.93 

3.69 

93.91 

4.10 

93.89 

4.51 

91 

S5 

94.94 

3.32 

94.93 

3.73 

94.91 

4.14 

94.89 

4.56 

95 

96 

95.94 

3.35 

95.93 

3.77 

95.91 

4.19 

95.89 

4.61 

96 

97 

96.94 

3.39 

96.93 

3.81 

96.91 

4.23 

96.89 

4.65 

97 

98 

97.94 

3.42 

97.92 

3.85 

97.91 

4.27 

97.89 

4.70 

98 

99 

98.94 

3.46 

98.92 

3.89 

98.91 

4.32 

98.89 

4.75 

99 

100 

99.94 

3.49 

99.92 

3.93 

99.91 

4.36 

99.88 

4.80 

100 

o> 

p 

Dep. 

Lac. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

§ 

c 

1 

1 

88  Deg. 

87|  Deg. 

87£  Deg. 

87*  Deg. 

• 
P 

TRAVERSE   TABLE. 


c 

3  Deg. 

31  Deg. 

3£  Deg. 

3|  Deg. 

O 

w" 

1 

(0 

5 

a 

o 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.00 

0.05 

1.  00 

0.06 

Too" 

0.06 

1.00 

0.06 

I 

2 

2.00 

0.10 

2.00 

0.11 

2.00 

0.12 

2.00 

0.13 

2 

3 

3.00 

0.16 

3.00 

0.17 

2.99 

0.18 

2.99 

0.20 

3 

4 

3.99 

0.21 

3.99 

0.23 

3.99 

0.24 

3.99 

0.26 

4 

5 

4.99 

0.26 

4.99 

0.28 

4.99 

0.31 

4.99 

0.33 

5 

6 

5.99 

0.31 

5.99 

0.34 

5.99 

0.37 

5.99 

0.39 

6 

7 

6.99 

0.37 

6.99 

0.40 

6.99 

0.43 

6.99 

0.46 

7 

8 

7.99 

0*42 

7.99 

0.45 

7.99 

0.49 

7.98 

0.52 

8 

9 

8.99 

0.47 

8.99 

0.51 

8.98 

0.55 

8.98 

0.59 

9 

10 

9.99 

0.52 

9.98 

0.57 

9.98 

0.61 

9.98 

0.65 

10 

11 

10.98 

0.58 

10.98 

0.62 

10.98 

0.67 

10.98 

0.72 

11 

12 

11.98 

0.63 

11.98 

0.68 

11.98 

0.73 

11.97 

0.78 

12 

13 

12.98 

0.68 

12.98 

0.73 

12.98 

0.79 

12.97 

0.85 

13 

14 

13.98 

0.73 

13.98 

0.79 

13.97 

0.85 

13.97 

0.92 

14 

15 

14.98 

0.79 

14.98 

0.85 

14.97 

0.92 

14.97 

0.98 

15 

16 

15.98 

0.84 

15.97 

0.91 

15.97 

0.98 

15.97 

.05 

16 

17 

16.98 

0.89 

16.97 

0.96 

16.97 

1.04 

16.96 

.11 

17 

18 

17.98 

0.94 

17.97 

1.02 

17.97 

1.10 

17.96 

.18 

18 

19 

18.98 

0.99 

18.97 

1.08 

18.96 

1.16 

18.96 

.24 

19 

20 

19.97 

1.05 

19.97 

1  .  r.i 

19.96 

1.22 

19.98 

.31 

20 

21 

20.97 

.10 

20.97 

.19 

20.96 

1.28 

20.96 

.37 

21 

22 

21.97 

.15 

21.96 

.25 

21.96 

1.34 

21.95 

.44 

22 

23 

22,97 

.20 

22.96 

.30 

22.96 

1.40 

22.95 

.50 

23 

24 

23.97 

.26 

23.96 

.36 

23.96 

1.47. 

23.95 

.57 

24 

25 

24.97 

.31 

24.96 

.42 

24.95 

1.53 

24.95 

.64 

25 

26 

25.96 

.36 

25.96 

.47 

25.95 

1.59 

25  .  94 

.70 

26 

27 

26.96 

.41 

26.96 

.53 

26.95 

1.65 

26.94 

.77 

27 

28 

27.96 

.47 

27.95 

.59 

27.95 

1.71 

27.94 

.83 

28 

29 

28.96 

.52 

28.95 

.64 

28.95 

1.77 

28.94 

1.90 

29 

30 

29.96 

.57 

29  .  95 

.  .  70 

29.94 

1.83 

29.94 

1.96 

30 

31 

30.96 

.62 

30.95 

.76 

30.94 

1.89* 

30.93 

2.03 

31 

32 

31.96 

.67 

31.95 

.81 

•31.94 

1.95 

31.93 

2.09 

32 

33 

32.95 

.73 

32.95 

.87 

32.94 

2.0J 

32.93 

2.16 

33 

34 

33.95 

.78 

33.95 

.93 

33.94 

!2.08 

33.93 

2.22 

34 

35 

34.95 

.83 

34.94 

.98 

34.93 

2.11 

34.92 

2.29 

35 

36 

35.95 

.88 

35.94 

2.04 

35  .  93 

2.20 

35.92 

2.35 

36 

37 

36  .  95 

.94 

36.94 

2.10 

36.93 

2.26 

36.92 

2.42 

37 

38 

37.95 

.99 

37.94 

2.15 

37.93 

2.32 

37.92 

2.49 

38 

39 

38.95 

2.04 

38  .  94 

2.21 

38.93 

2.38 

38  .  92 

2.55 

39 

40 

39.95 

2«09 

39.94 

2.27 

39.93 

2.44 

39.91 

2.62 

40 

41 

40.94 

2.15 

40  .  93 

2.32 

40.92 

2.50 

40.91 

2.68 

~41 

42 

41.  &4 

2.20 

41  .  93 

2.38 

41.92 

2.56 

41.91 

2.75 

42 

43 

42.94 

2.25 

42.93 

2.44 

42.92 

2.63 

42.91 

2.81 

43 

44 

43.94 

2.30 

43.93 

2.49 

43  .  92 

2.69 

43.91 

2.88 

44 

45 

44.94 

2.36 

44.93 

2.55 

44.92 

2.75 

44.90 

2.94 

45 

46 

45.94 

2.41 

45.93 

2.61 

45.91 

2.81 

45.90 

3.01 

46 

47 

46.94 

2.46 

46  .  92 

2.66 

46.91 

2.87 

46.90 

3.07 

47. 

48 

47.93 

2.51 

47.92 

2.72 

47.91 

2.93 

47.90 

3.14 

48 

49 

48.93 

2.56 

48.92 

2.78 

48.91 

2.99i  48.90 

3.20 

49 

50 

49.93 

2.62 

49.92 

2.83 

49.91 

3.05! 

49.89 

3.27 

50 

OJ 

o 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

5 

87  Deg. 

86J  Deg. 

86A  Deg. 

861  Deg. 

CO 

.5 
Q 

TRAVERSE   TABLE. 


o 

3  Deg. 

3i  Deg. 

3±  Deg. 

3|  Deg. 

2 

1 

S- 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

I 

51 

50.93 

2.67 

50.92 

2.89 

50.90 

3.11 

50.89 

3.34 

51 

52 

51.93 

2.72 

51.92 

2.95 

51.90 

3.17 

51.89 

3.40J  52 

53 

52.93 

2.77 

52.91 

3.00 

52.90 

3.24 

52.89 

3/47 

53 

54 

53.93 

2.83 

53.91 

3.06 

53.90 

3.30 

53.88 

3.53 

54 

55 

54.92 

2.88 

54.91 

3.12 

54.90 

3.36 

54.88 

3.60 

55 

56 

55.92 

2.93 

55.91 

3.17 

55.90 

3.42 

55.88 

3.66 

56 

57 

56.92 

2.98 

56.'91 

3.23 

56.89 

3.48 

56.88 

3  73 

57 

58 

57.92 

3.04 

57.91 

3.29 

57.89 

3.54 

57.88 

3.79 

58 

59 

58.92 

3.09 

58.91 

3.34 

58.89 

3.60 

58.87 

3.86 

59 

60 

59.92 

3.14 

59.90 

3.40 

59.89 

3.66 

59.87 

3.92 

60 

61 

60.92 

3.19 

60.90 

3.46 

60.89 

3.72 

60.87 

3.99 

61 

62 

61.92 

3.24 

61.90 

3.51 

61.88 

3.79 

61.87 

4.05 

62 

63 

62.91 

3.30 

62.90 

3.57 

62.88 

3.85 

62.87 

4.12 

63 

64 

63.91 

3.35 

63.90 

3.63 

63.88 

3.91 

63.86 

4.19 

64 

65 

64.91 

3.40 

64.90 

3.69 

64.88 

3.97 

64.86 

4.25 

65 

66 

65.91 

3.45 

65.89 

3.74 

65.88 

4.03 

65.86 

4.32 

66 

67 

66.91 

3.51 

66.89 

3.80 

66.88 

4  09 

66.86 

4.38 

67 

68 

67.91 

3.56 

67.89 

3.86 

67.87 

4.15 

67.85 

4.45 

68 

69 

68.91 

3.61 

68.89 

3.91 

68.87 

4.21 

68.85 

4.51 

69 

70 

69.90 

3.66 

69.89 

3.97 

69.87 

4.27 

69.85 

4.58 

70 

71. 

70.90 

3.72 

70.89 

4.03 

70.87i  4.33 

70.85 

4.64 

71 

72 

71.90 

3.77 

71.88 

4.08 

71.87 

4.40 

71.85 

4.71 

72 

73 

72.90 

3.82 

72.88 

4.14 

72.86 

4.46 

72.84 

4.77 

73 

74 

73.90 

3.87 

73.88 

4.20 

73.86 

4.52 

73.84 

4.84 

74 

75 

74.90 

3.93 

74.88 

4.25 

74.86 

4.58 

74.84 

4.91 

75 

76 

75.90 

3.98 

75.88 

4.31 

75-86 

4.64 

75.84 

4.97 

76 

77 

76.89 

4.03 

76.88 

4.37 

76.86 

4.70 

76.84 

5.04 

77 

78 

77.89 

4.08 

77.87 

4.42 

77.85 

4.76 

77.83 

5.10 

78 

79 

78.89 

4.13 

78.87 

4.48 

78.85 

4.82 

78.83 

5.17 

79 

80 

79.89 

4.19 

79.87 

4.54 

79.85 

4.88 

79.83 

5.23 

80 

81 

80.89 

4.24 

80.87 

4.59 

80.85* 

4.94 

80.83 

5.30 

81 

82 

81.89 

4.29 

81.87 

4.65 

81.85 

5.01 

81.82 

5.38 

82 

83 

82.89 

4.34 

82.87 

4.71 

82.85 

5.07 

82.82 

5.43 

83 

84 

83.88 

4.40 

83.86 

4.76 

83.84 

5.13 

83.82 

5.49 

84 

85 

84.88 

4.45 

84.86 

4.82 

84.84 

5.19 

84.82 

5.56 

85 

86 

85.88 

4.50 

85.86 

4.88 

85.84 

5.25 

85.82 

5.62 

86 

87 

86.88 

4.55 

86.86 

4.93 

86.84 

5.31 

86.81 

5.69 

87 

88 

87.88 

.4.61 

87.86 

4.99 

87.84 

5.37 

87.81 

5.7Q 

88 

89 

88.88 

4.66 

88.86 

5.05 

88.83 

5.43 

88.81 

5.82 

89 

90 

89.88 

4.71 

89.86 

5.10 

89.83 

5.49 

89.81 

5.89 

90 

91 

90.88 

4.76 

90.85 

'5.16 

90.83 

5.56 

90.81 

5.95 

91 

92 

91.87 

4.81 

91.85 

5.22 

91.83 

5.62 

91.80 

6.02 

92 

93 

92.87 

4.87 

92.85 

5.27 

92.83 

5.68 

92.80 

6.08 

93 

94 

93.87 

4.92 

93.85 

5.33 

93.82 

5.74 

93.80 

6.15 

94 

95 

94.87 

4.97 

94.85 

5.39 

94.82 

5.80 

94.80 

6.21 

95 

96 

95.87 

5.02 

95.85 

5.  '44 

95.82 

5.86 

95.79 

6.28 

96 

97 

96.87 

5.08 

96.84 

5.50 

96.82 

5.92 

96.79 

6.34 

97 

98 

97.87 

5.13 

97.84 

5.56 

97.82 

5.98 

97.79 

6.41 

98 

99 

98.86 

5.18 

98.84 

5.61 

98.82 

6.04 

98.79 

6.47 

99 

100 

99.86 

5.23 

99.84 

5.67 

99.81 

6.10 

99.79 

6.54 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

C 

87  Deg. 

86|  Deg. 

8G|  Deg. 

86i  Deg. 

H 

1 

10 


TRAVERSE    TABLE. 


b 

4  Deg. 

4|  Deg. 

4-'  Deg. 

4|  Deg. 

O 
5 

i 

? 

a 
5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 
P 

i 

1.00 

0.07 

1.00 

0.07 

1.00 

0..08 

1.00 

0.08 

1 

2 

2.00 

0.14 

1.99 

0.15 

1.99 

0.16 

1.99 

0.17 

2 

3 

2.99 

0.21 

2.99 

0.22 

2.99 

0.24 

2.99 

0.25 

3 

4 

3.99 

0.28 

3.99 

0.30 

3.99 

0.31 

3.98 

0.33 

4 

5 

4.99 

0.35 

4.99 

0.37 

4.98 

0.39 

4.98 

0.41 

5 

6 

5.99 

0.42 

5.98 

0.44 

5.98 

0.47 

5.98 

0.50 

6 

7 

6.98 

0.49 

6.98 

0.52 

6.98 

0.55 

6.97 

0.58 

7 

8 

7.98 

0.56 

7.98 

0.59 

7.98 

0.63 

7.97 

0.66 

8 

9 

8.98 

0.63 

8.98 

0.67 

8.97 

0.7J 

8.97 

0.75 

9 

10 

9.98 

0.70 

9.97 

0.74 

9.97 

0.78 

9.97 

0.83 

10 

11 

10.97 

0.77 

10.97 

6.82  i 

10.97 

0.86 

10.96 

O.S1 

11 

12 

11.97 

0.84 

11.97 

0.89 

11.96 

0.94 

11.96 

0.99 

12 

13 

12.97 

0.91 

12.96 

0.96 

12.96 

1.02 

12.96 

.08 

13 

14 

13.97 

0.98 

13.96 

1.C4 

13.96 

1.10 

3.95 

.16 

14 

15 

14.96 

1.05 

14.96 

1.11 

14.95 

l.lg 

14.95 

.24 

15 

16 

15.96 

1.12 

15.96 

1.19 

15.95 

1.26 

15.95 

.32 

16 

17 

16.96 

1.19 

16.95 

1.26 

16.95 

1.33 

16.94 

.41 

17 

18 

17.96 

1.26 

17.95 

1.33 

17.94 

1.41 

17.94 

.49 

18 

19 

18.95 

1.33 

18.95 

1.40 

18.94 

1.49 

18.93 

1.57 

19 

20 

19.95 

1.40 

19.95 

1.48 

19.94 

1.57 

19.93 

1.66 

20 

21 

20.95 

1.46 

20.94 

1.56 

20.94 

1.651 

20.93 

1.74 

21 

22 

21.95 

1.53 

21.94 

1.63 

21.93 

1.73 

21.92 

1.82 

22 

23 

22.94 

1.60 

22.94 

1.70 

22.93 

1.80 

22.92 

1.90 

23 

24 

23.94 

•1.67 

23.93 

1.78 

23.93 

1.88 

23.92 

1.99 

24 

25 

24.94 

1.74 

24.93 

1.85 

24.92 

1.96 

24.91 

2.07 

25 

26 

25.94 

1.81 

25.93 

1.93 

25.92 

2.04 

25.91 

2.15 

26 

27 

26.93 

1.88 

20.93 

2.00 

26.92 

2.12 

26.91 

2.24 

27 

28 

27.93 

1.95 

27.92 

2.08 

27.91 

2.20 

27.90 

2.32 

28 

29 

28.93 

2.02 

28.92 

2.15 

28.91 

2.28 

28.90 

2.40 

29 

.30 

29.93 

2.09 

29.92 

2.22 

29.91 

2.35 

29.90 

2.48 

30 

31 

30.92 

2.16 

30.91 

*2.30 

30.90 

2.43 

30.89 

2.57 

31 

32 

31.92 

2.23 

31.91 

2.37 

31.90 

2;51 

31.89 

2.65 

32 

33 

32.92 

2.30 

32.91 

2.45 

32.90 

2.59 

32.89 

2.73 

33 

34 

3-3.92 

2.37 

33.91 

2.52 

33.90 

2.67 

33.88 

2.82 

34 

35 

34.91 

2.44 

34.90 

2.59 

34.89 

2.75 

34.88 

2.90 

35 

36 

35.91 

2.51 

35.90 

2.67 

35.89 

2.82 

35.88 

2.98 

36 

37 

36.91 

2.58 

36.90 

2.74 

36.89 

2.90 

36.87 

3.06 

37 

38 

^7.  91 

2.65 

37.90 

2.82 

37.88 

2.98 

37.87 

3.15 

38 

39 

38.90 

2.72 

38.89 

2.89 

38.88 

3.06 

38.87 

3.23 

39 

40 

39.90 

2.79 

39.89 

2.96 

39.88 

3.14 

39.86 

3.31 

40 

41 

40.90 

2.86 

40.89 

3.04 

40.87 

3.22 

40.86 

3.40 

41 

42 

41.90 

2.93 

41.88 

3.11 

41.87 

3.30 

41.86 

3.4S 

42 

43 

42.90 

3.00 

42.88 

3.19 

42.87 

3.37 

42.85 

8.56 

43 

44 

43.89 

3.07 

43.88 

3.26 

43.86 

3.45 

43.85 

3.64 

44 

45 

41.89 

3.14 

44.88 

3.33 

44.86 

3.53 

44.85 

3.73 

45 

46 

45.89 

3.21 

45.87 

3.41 

45.86 

3.61 

45.84 

3.81 

46 

47 

46.89 

3.28 

46.87 

3.48 

46.86 

3.69 

46.84 

3.89 

47 

48 

47.88 

3.35 

47.87 

3.56 

47.85 

3.77 

47.84 

3.97 

48 

49 

48.88 

3.42 

48.87 

3.63 

48.85 

3.84 

48.83 

4.06 

49 

50 

49.881  3.49 

49.86 

3.71 

49.85 

3.92 

49.83 

4.14 

50 

Q> 

O 

c 

Dep.  |  Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

CJ 
O 

S3 

• 

1C 

3 

86  Deg. 

85|  Deg. 

85£  Deg. 

85J  Deg. 

rt 

Q 

TBAVERSE    TABLE. 


11 


o 

53' 
? 

4  Deg. 

4i  Deg. 

4^  Deg. 

4JDeg. 

g 

a 

n 
P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

51 

50.88 

3.56 

50.86 

3.78 

50.84 

4.00 

50.82 

4.22 

51 

52 

51.87 

3.63 

51.86 

3.85 

51.84 

4.08 

51.82 

4.31 

52 

53 

52.87 

3.70 

52.85 

3.93 

52.84 

4.16 

52.82 

4.39 

53 

'  54 

53.87 

3.77 

53.85 

4.00 

53.83 

4.24 

53.81 

4.47 

54 

55 

54.87 

3.84 

54.85 

4.08 

54.83 

4.32 

54.81 

4.55 

55 

56 

55.86 

3.91 

55.85 

4.15 

55.83 

4.39 

55.81 

4.64 

56 

57 

56.86 

3.98 

56.84 

4.22 

56.82 

4.47 

56.80 

4.72 

57 

58 

57.86 

4.05 

57.84 

4.30 

57.82 

4.55 

57.80 

4.80 

58 

59 

58.86 

4.12 

58.84 

4.37 

58.82 

4.63 

58.80 

4.89 

59 

60 

59.85 

4.19 

59.84 

4.45 

59.82 

4.71 

59.79 

4.97 

60 

61 

60.85 

4.26 

60.83 

4.52 

60.81 

4.79 

60.79 

5.05 

61 

62 

61.85 

4.32 

61.83 

4.59 

61.81 

4.86 

61.79 

5.13 

62 

63 

62.85 

4.39 

62.83 

4.67 

62.81 

4.94 

62.78 

5.22 

63 

64 

63.84 

4.46 

63.82 

4.74 

63.80 

5.02 

63.78 

5.30 

64 

65 

64.84 

4.53 

64.82 

4.82 

64.80 

5.10 

64.78 

5.38 

65 

66 

65.84 

4.60 

65.82 

4.89 

65.80 

5.18 

65.77 

5.47 

66 

67 

66.84 

4.67 

66.82 

4.97 

66.79 

5.26 

66.77 

5.55 

67 

68 

67.83 

4.  ,74 

67.81 

5.04 

67.79 

5.34 

67.77 

5.63 

68 

69 

68.83 

4.81 

68.81 

5.11 

68.79 

5.41 

68.76 

5.71 

69 

70 

69.83 

4.88 

69.81 

5.19 

69.78 

5.49 

69,76 

5.80 

70 

71 

70.83 

4.95 

70.80 

5.26 

70.78 

5.57 

70.76 

5.88 

71 

72 

71.82 

5.02 

71.80 

5.34 

71.78 

5.65 

71.75 

5.96 

72 

73 

72.82 

5.09 

72.80 

5.41 

72.77 

5.73 

72.75 

6.04 

73 

74 

73.82 

5.16 

73.80 

5.48 

73.77 

5.81 

73.75 

6.13 

74 

75 

74.82 

5.23 

74.79 

5.56 

74.77 

5.88 

74.74 

6.21 

75 

76 

75.81 

5.30 

75.79 

5.63 

75.77 

5.96 

75.74 

6.29 

76 

77 

76.81 

5.37 

76.79 

5.71 

76.76 

6.04 

76.74 

6.38 

77 

78 

77.81 

5.44 

77.79 

5.78 

77.76 

6.12 

77.73 

6.46 

78 

79 

78.81 

5.51 

78.78 

5.85 

78.76 

6.20 

78.73 

6.54 

79 

80 

79.81 

5.58 

79.78 

5.93 

79.75 

6.28 

79.73 

6.62 

80 

81 

80.80 

5.65 

80.78 

6.00 

80.75 

6.36 

80.72 

6.71 

81 

82 

81.80 

5.72 

81.78 

6.08 

81.75 

6.43 

81.72 

6.79 

82 

83 

82.80 

5.79 

82.77 

6.15 

82.74 

6.51 

82.71 

6.87 

83 

84 

83.80 

5.86 

83.77 

6.23 

83.74 

6.59 

83.71 

6.96 

84 

85 

84.79 

5.93 

84.77 

6.30 

84.74 

6.67 

84.71 

7.04 

85 

86 

85.79 

6.00 

85.76 

6.3? 

85.73 

6.75 

85.70 

7.12 

86 

87 

86.79 

6.07 

86.76 

6.45 

86.73 

6.83 

86.70 

7.20 

87 

88 

87.79 

6.14 

87.76 

6.52 

87.73 

6.90 

87.70 

7.29 

83 

89 

88.78 

6.21 

88.76 

6.60 

88.73 

6.98 

88.70 

7.37 

89 

90 

89.78 

6.28 

89.75 

6.67 

89.72 

7.06 

89.69 

7.45 

90 

91 

90.78 

6.35 

90.75 

6.74! 

90.72 

7.14 

90.69 

7.54 

91 

92 

91.78 

6.42 

91.75 

6.82 

91.72 

7.22 

91.68 

7.62 

92 

93 

92.77 

6.49 

92.74 

6.89 

92.71 

7.30 

92.68 

7.70 

93 

94 

93.77 

6.56 

93.74 

6.97 

93.71 

7.38 

93.68 

7.78 

94 

95 

94.77 

6.63 

94.74 

7.04 

94.71 

7.45 

94.67 

7.87 

95 

96 

95.77 

6.70 

95.74 

7.11 

95.70 

7.53 

95.67 

7.95 

96 

97 

96.76 

6.77 

96.73 

7.19 

96.70 

7.61 

96.67 

8.03 

97 

98 

97.76 

6.84 

97.73 

7.26 

97.70 

7.69 

97.66 

8.12 

98- 

99 

98.76 

6.91 

98.73 

7.34 

98.69 

7.77 

98.66 

8.20 

99 

100 

99.76   6.98 

99.73 

7.41 

99.69 

7.85 

99.66 

8.28 

100 

0 

u 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

§ 

•  C 

c3 
• 

3 

86  Deg. 

85|  Deg. 

85£  Deg. 

85i  Deg. 

c3 

.3 

b 

K 


12 


TRAVERSE   TABLE. 


g 

en* 

P 

5  Deg. 

5}  Deg, 

51  Deg. 

6J  Deg. 

5 

w* 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

i 

1.00 

0.09 

1.00 

0.09 

1.00 

0.10 

0.99 

0.10 

1 

2 

1.99 

0.17 

1.99 

0.18 

1.99 

0.19 

1.99 

0.20 

2 

3 

2.99 

0.26 

2.99 

0.27 

2.99 

0.29 

2.98 

0.30 

3 

4 

3.98 

0.35 

3.98 

0.37 

3.98 

0.38 

3.98 

0.40 

4 

5 

4.98 

0.44 

4.98 

0.46 

4.98 

0.48 

4.97 

0.50 

5 

6 

5.98 

0.52 

5.97 

0.55 

5.97 

0.58 

5.97 

0.60 

6 

7 

6.97 

0.61 

6.97 

0.64 

6.97 

0.67 

6.96 

0.70 

7 

8 

7.97 

0.70 

7.97 

0.73 

7.96 

0.76 

7.96 

0.80 

8 

9 

8.97 

0.78 

8,96 

0.82 

8  96 

0.86 

8.95 

0.90 

9 

10 

9.96 

0.87 

9.96 

0.92 

9.95 

0:96 

9.95 

1.00 

10 

11 

10.96 

0.96 

10.95 

.01 

10.95 

.05 

10.94 

1.10 

11 

12 

11.95 

1.05 

11.95 

.10 

11.94 

.15 

11.94 

1.20 

12 

13 

12.95 

1.13 

12.95 

.19 

12.94 

.25 

12.93 

1.30 

13 

14 

13.95 

1.22 

13.94 

.28 

13.94 

.34 

13.93 

1.40 

14 

15 

14.94 

1.31 

14.94 

.37 

14.93 

.44 

14.92 

1.50 

15 

16 

15.94 

1.39 

15.93 

.46 

15.93 

.53 

15.92 

1.60 

16 

17 

16.94 

1.48 

16.93 

.56 

16.92 

.63 

16.91 

1.70 

17 

13 

17.93 

1.57 

17.92 

.65 

17.92 

.73 

17.91 

1.80 

18 

19 

18.93 

1.66 

18.92 

.74 

18.91 

.82 

18.90 

1.90 

19 

20 

19.92 

1.74 

19.92 

.83 

19.91 

1.92 

19.90 

2.00 

20 

21 

20.92 

1.83J 

20.91 

1.92 

20.90 

2.01  1 

20.89 

2.10 

21 

22 

21.92 

1.92 

21.91 

2.01 

21.90 

2.11 

21.89 

2.20 

22 

23 

22.91 

2.00 

22.90 

2.10 

22.89 

2.20 

22.88 

2.30 

23 

24 

23.91 

2.09 

23.90 

2.20 

23.89 

2.30 

23.88 

2.40 

24 

25 

24.90 

2.18 

24.90 

2.29 

24.88 

2.40 

24.87 

2.50 

25 

26 

25.90 

2.27 

25.89 

2.38 

25.88 

2.49 

25.87 

2.60 

26 

27 

26.90 

2.35 

26.89 

2.47 

26.88 

2.59 

26.86 

2.71 

27 

28 

27.89 

2.44 

27.88 

2.56 

27.87 

2.68 

27.86 

2.81 

28 

29 

28.89 

2.53 

28.88 

2.65 

28.87 

2.78 

28.85 

2.91 

29 

30 

29.89 

2.61 

29.87 

2.75 

29.86 

2.88 

29.85 

3.01 

30 

31 

30.88 

2.70 

30.87 

2.84 

30.86 

2.97 

30.84 

3.11 

31 

32 

31.88 

2.79 

31.87 

2.93 

31.85 

3.07 

31.84 

3.  -21 

32 

33 

32.87 

2.88 

32.86 

3.02 

32.85 

3.16 

32.83 

3.31 

33 

34 

33.87 

2.96 

33.86 

3.11 

33.84 

3.26 

33.83 

3.41 

34 

35 

34.87 

3.05 

34.85 

3.20 

34.84 

3.35 

34.82 

3.51 

35 

36 

35.86 

3.14 

35.85 

3.29 

35.83 

3.45 

35.82 

3.61 

36 

37 

36.86 

3.22 

36.84 

3.39 

36.83 

3.55 

36.81 

3.71 

37 

38 

37.86 

3.31 

37.84 

3.48 

37.83 

3.64 

37.81 

3.81 

38 

39 

38.85 

3.40 

38.84 

3.57 

38.82 

3.74 

38.80 

3.91 

39 

40 

39.85 

3.49 

39.83 

3.66 

39.82 

3.83 

39.80 

4.01 

40 

41 

40.84 

3.57 

40.83 

3.75 

40.81 

3.93 

40.79 

4.J.1 

41 

42 

41.84 

3.66 

41.82 

3.84 

41.81 

4.03 

41.79 

4.21 

42 

43 

42.84 

3.75 

42.82 

3.93 

42.80 

4.12 

42.78 

4.31 

43 

44 

43.83 

3.83 

43.82 

4.03 

43.80 

4.22 

43.78 

4.41 

44 

45 

44.83 

3.92 

44.81 

4.12 

44.79 

4.31 

44.77 

4.51 

45 

46 

45.82 

4.01 

45.81 

4.21 

45.79 

4.41 

45.77 

4.61 

46 

47 

46.82 

4.10 

46.80 

4.30 

46.78 

4.50 

46.76 

4.71 

47 

48 

47.82 

4.18 

47.80 

4.39 

47.78 

4.60 

47.76 

4.81 

48 

49 

48.81 

4.27 

48.79 

4.48 

48.77 

4.70 

48.75 

4.91 

49 

50 

49,81 

4.36 

49,79 

4.58 

49.77 

4.79 

49.75 

5.01 

50 

I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

09 

O 

c 

a 

c3 

00 

s 

85  Deg. 

84J  Deg. 

841  Deg. 

84i  Deg. 

.a 

b 

TRAVERSE    TABLE. 


13 


G 

5  Deg. 

5i  Deg. 

H  De«- 

5|  Deg. 

D 

£' 

p 

1' 

3 
n 
? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

51 

50.81 

4.44 

50.79 

4.67 

50.77 

4.89 

50.74 

5.11 

51 

52 

51.80 

4.53 

51.78 

4.76 

51.76 

4.98 

51.74 

5.21 

52 

53 

52.80 

4.62 

52.78 

4.85 

62.76 

5.08 

52.73 

5.31 

53 

54 

53.79 

4.71 

53.77 

4.94 

53.75 

5.18 

53.73 

5.41 

54 

55 

54.79 

4.79 

54.77 

5.03 

54.75 

5.27 

54.72 

5.51 

55 

56 

55.79 

4.88 

55.77 

5.12 

55.74 

5.37 

55.72 

5.61 

56 

57 

56.78 

4.97 

56.76 

5.22 

56.74 

5.46 

56.71 

5.71 

57 

58 

57.78 

5.06 

57.76 

5.31 

57.73 

5.56 

57.71 

5.81 

58 

59 

58.78 

5.14 

58.75 

5.40 

58.73 

5.65 

58.70 

5.91 

59 

60 

59.77 

5.23 

59.75 

5.49 

59.72 

5.75 

59.70 

6.01 

60 

61 

60.77 

5.32 

60.74 

5.58 

60.72 

5.85 

60.69 

6.11 

61 

62 

61.76 

5.40 

61.74 

5.67 

61.71 

5.94 

61.69 

6.21 

62 

63 

62.76 

5.49 

62.74 

5.76 

62.71 

6.04 

62.68 

6.31 

63 

64 

63.76 

5.58 

63.73 

5.86 

63.71 

6.13 

63.68 

6.41 

64 

65 

64.75 

5.67 

64.73 

5.95 

64.70 

6.23 

64.67 

6.51 

65 

66 

65.75 

5.75 

65.72 

6.04 

65.70 

6.33 

65.67 

6.61 

66 

67 

66.75 

5.84 

66.72 

6.13 

66.69 

6.42 

66.66 

6.71 

67 

63 

67.74 

5.93 

67.71 

6.22 

67.69 

6.52 

67.66 

6.81 

68 

69 

68.74 

6.01 

68.71 

6.31 

68.68 

6.61 

68.65 

6.91 

69 

70 

69.73 

6.10 

69.71 

6.41 

69.68 

6.71 

69.65 

7,01 

70 

71 

70.73 

6.19 

70.70 

6.50 

70.67 

6.81 

70.64 

7.11 

71 

72 

71.73 

6.28 

71.70 

6.59 

71.67 

6.90 

71.64 

7.21 

72 

73 

72.72 

6.36 

72.69 

6.68 

72.66 

7.00 

72.63 

7.31 

73 

74 

73.72 

6.45 

73.69 

6.77 

73.66 

7.09 

73.63 

7.41 

74 

75 

74.71 

6.54 

74.69 

6.86 

74.65 

7.19 

74.62 

7.51 

75 

76 

75.71 

6.62 

75.68 

6.95 

75.65 

7.28 

75.62 

7.61 

76 

77 

76.71 

6.71 

76.68 

7.05 

76.65 

7.38 

76.61 

7.71 

77 

78 

77.70 

6.80 

77.67 

7.14 

77.64 

7.48 

77.61 

7.81 

78 

79 

78.70 

6.89 

78.67 

7.23 

78.64 

7.57 

78.60 

7.91 

79 

80 

79.70 

6.97 

79.66 

7.32 

79.63 

7.67 

79.60 

8.02 

80 

81 

80.69 

7.06 

80.66 

7.41 

80.63 

7.76 

80.59 

8.12 

81 

82 

81.69 

7.15 

81.66 

7.50 

81.62 

7.86 

81.59 

8.22 

82 

83 

82.68 

7.23 

82.65 

7.59 

82.62 

7.96 

82.58 

8.32 

83 

84 

83.68 

7.32 

83.65 

7.69 

83.61 

8.05 

83.58 

8.42 

84 

85 

84.68 

7.41 

84.64 

7.78 

84.61 

8.15 

84.57 

8.52 

85 

86 

85.67 

7.50 

85.64 

7.87 

85.60 

8.24 

85.57 

8.62 

86 

87 

86.67 

7.58 

86.64 

7.96 

86.60 

8.34 

86.56 

8.72 

87 

88 

87.67 

7.67 

87.63 

8.05 

87.59 

8.43 

87.56 

8.82 

88 

89 

as.  66 

7.76 

88.63 

8.14 

88.59 

8.53 

88.55 

8.92 

89 

90 

80.66 

7.84 

89.62 

8.24 

89.59 

8.63 

89.55 

9.02 

90 

91 

90.65 

7.93 

90.62 

8.33 

90.58 

8.72 

90.54 

9.12 

91 

92 

91.65 

8.02 

91.61 

8.42 

91.58 

8.82 

91.54 

9.22 

92 

93 

92.65 

8.11 

92.61 

8.51 

92.57 

8.91 

92.53 

9.32 

93 

94 

93.64 

8.19 

93.61 

8.60 

93.57 

9.01 

93.53 

9.42 

94 

95 

94.64 

8.28 

94.60 

8.69 

94.56 

9.11 

94.52 

9.52 

95 

96 

95.63 

8.37 

95.60 

8.78 

95.56 

9.20 

95.52 

9.62 

96 

97 

96.63 

8.45 

96.59 

8.88 

96.55 

9.30 

96.51 

9.72 

97 

98 

97.63 

8.54 

97.59 

8.97 

97.55 

9.39 

97.51 

9.82 

98 

99 

98.62 

8.63 

98.59 

9.06 

98.54 

9.49 

98.50 

9.92 

99 

100 

99.62 

8.72 

99.58 

9.15 

99.54 

9.58 

99.50 

10.02 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

§ 

I 

85  Deg. 

84J  Deg. 

84£  Deg. 

84*  Deg. 

£ 

5 

14 


TRAVERSE    TABLE. 


o 

6  I 

)eg. 

m 

)eg. 

6^1 

)eg. 

6|  I 

)eg. 

O 

I 

P* 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 

n 

CD 

1 

0.99 

0.10 

0.99 

0.11 

0.99 

0.11 

0.99 

~"07l2~ 

1 

2 

1.99 

0.21 

1.99 

0.22 

1.99 

0.23 

1.99 

0.24 

2 

3 

2.98 

0.31 

2.98 

0.33 

2.98 

0.34 

2.98 

0.35 

3 

4 

3.98 

0.41 

3.98 

0.44 

3.97 

0.45 

3.97 

0.47 

4 

5 

4.97 

0.52 

4.97 

0.54 

4.97 

0.57 

4.97 

0.59 

5 

6 

5.97 

0.63 

5.96 

0.65 

5.96 

0.68 

5.96 

0.71 

6 

7 

6.96 

0.73 

6.96 

0.76 

6.96 

0.79 

6.95 

0.82 

7 

8 

7.96 

0.84 

7.95 

0.87 

7.95 

0.91 

7.94 

0.94 

8 

9 

8.95 

0.94 

8.95 

0.98 

8.94 

.02 

8.94 

1.06 

9 

10 

9.95 

1.05 

9.94 

1.09 

9.94 

.13 

9.93 

1.18 

10 

11 

10.94 

1.15 

10.93 

.20 

10.93 

.25 

10.92 

1.29 

11 

12 

11.93 

1.25 

11.93 

.31 

11.92 

.36 

11.92 

1.41 

12 

13 

12.93 

1.36 

12.92 

.42 

12.92 

.47 

12.91 

1.53 

13 

14 

13.92 

1.46 

13.92 

.52 

13.91 

.59 

13.90 

1.65 

14 

15 

14.92 

1.57 

14.91 

.63 

14.90 

.70 

14.90 

1.76 

15 

16 

15.91 

1.67 

15.90 

.74 

15.90 

1.81 

15.89 

1.88 

16 

17 

16.91 

1.78 

16.90 

.85 

16.89 

1.92 

16.88 

2.00 

17 

18 

17.90 

1.88 

17.89 

1.96 

17.88 

2.04 

17.88 

2.12 

18 

19 

18.90 

1.99 

18.89 

2.07 

18.88 

2.15 

18.87 

2.23 

19 

20 

19.89 

2.09 

19.88 

2.18 

19.87 

2.26 

19.86 

2.35 

20 

21 

20.88 

2.20 

20.88 

2.29 

20.87 

2.38 

20.85 

2.47 

21 

22 

21.88 

2.30 

21.87 

2.40 

21.86 

2.49 

21.85 

2.59 

22 

23 

22.87 

2.40 

22.86 

2.50 

22.85 

2.60 

22.84 

2.70 

23 

24 

23.87 

2.51 

23.86 

2.61 

23.85 

2.72 

23.83 

2.82 

24 

25 

24.86 

2.61 

24.85 

2.72 

24.84 

2.83 

24.83 

2.94 

25 

26 

25.86 

2.72 

25.85 

2.83 

25.83 

2.94 

25.82 

3.06 

26 

27 

26.85 

2.82 

26.84 

2.94 

26.83 

3.06 

26.81 

3.17 

27 

28 

27.85 

2.93 

27.83 

3.05 

27.82 

3.17 

27.81 

3.29 

28 

29 

28.84 

3.03 

28.83 

3.16 

28.81 

3.28 

28.80 

3.41 

29 

30 

29.84 

3.14 

29.82 

3.27 

29.81 

3.40 

29.79 

3.53 

30 

31~ 

30.83 

3.24 

30.82 

3.37 

30.80 

3.51 

30.79 

3.64 

31 

32 

31.82 

3.34 

31.81 

3.48 

31.79 

3.62 

31.78 

3.76 

32 

33 

32.82 

3.45 

32.80 

3.59 

32.79 

3.74 

32.77 

3.88 

33 

34 

33.81 

3.55 

33.80 

3.70 

33.78 

3.85 

33.76 

4.00 

34 

35 

34.81 

3.66 

34.79 

3.81 

34.78 

3.96 

34.76 

4.11 

35 

36 

35.80 

3.76 

35.79 

3.92 

35.77 

4.08 

35,75 

4.23 

36 

37 

36.80 

3.87 

36.78 

4.03 

36.76 

4.19 

36.75 

4.35 

37 

38 

37.79 

3.97 

37.77 

4.14 

37.76 

4.30 

37.74 

4.47 

38 

39 

38.79 

4.08 

38.77 

4.25 

38.75 

4.41 

38.73 

4.58 

39 

40 

39.78 

4.18 

39.76 

4.35 

39.74 

4.53 

39.72 

4.70 

40 

41 

40.78 

4.29 

40.76 

4.46 

40.74 

4.64 

40.72 

4.82 

41 

42 

41.77 

4.39 

41.75 

4.5? 

41.73 

4.76 

41.71 

4.94 

42 

43 

42.76 

4.49 

42.74 

4.68 

42.72 

4.87 

42.70 

5.05 

43 

44 

43.76 

4.60 

43.74 

4.79 

43.72 

4.98 

43.70 

5.17 

44 

45 

44.75 

4.70 

44.73 

4.90 

44.71 

5.09 

44.69 

5.29 

45 

46 

45.75 

4.81 

45.73 

5.01 

45.70 

5.21 

45.68 

5.41 

46 

47 

46.74 

4.91 

46.72 

5.12 

46.70 

5.32 

46.67 

5.52 

47 

48 

47.74 

5.02 

47.71 

5.23 

47.69 

5.43 

47.67 

5.64 

48 

49 

48.73 

5.12 

48.71 

5.34 

48.69 

5.55 

48.66 

5.76 

49 

50 

49.73 

5.23 

49.70 

5.44 

49.68 

5.66 

49.65 

5.88 

50 

© 
1 

Dep.^ 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

QJ 
U 

c 

1 

G 

841 

Deg. 

83| 

Deg. 

831 

Deg. 

Deg. 

I 

TRAVERSE    TABLF.. 


15 


g 

6Degx 

6i  Deg. 

6i  Deg      6|  Deg. 

2 

So" 

1 

3 
o 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  1  Lat. 

Dep. 

3 
n 

51 

50.72 

5.33 

50.70 

5.55 

50.67 

5.77  50.65 

5.99 

~51 

52 

51.72 

5.44 

51.69 

5.66 

51.67 

5-89  51.64 

6.11 

52 

53 

52.71 

5.54 

52.68 

5.77 

52.66 

6.00  52.63 

6.23 

53 

54  53.70 

5.64 

53.68 

5.88 

53.65 

6-11 

53.63 

6.35 

54 

55 

54.70 

5.75 

54.67 

5.99 

54.65 

6-23 

54.62 

6.46 

55 

56 

55.69 

5.85 

55.67 

6.10 

55.64 

6-34 

55.61 

6.58 

56 

57 

56.69 

5.96 

56.66 

6.21 

56.63 

6-45 

56.60 

6.70 

57 

58  57.68 

6.06 

57.66 

6.31 

57.63 

6-57 

57.60 

6.82 

58 

59  58.68 

6.17 

58.65 

6.42 

58.62 

6.68 

58.59 

6.93 

59 

60 

59.67 

6.27 

59.64 

6.53 

59.61 

6.79 

59.58 

7.05 

60 

61 

60.67 

6.38 

60.64 

6.64 

60.61 

6.  91  ll  60.  58 

7.17 

61 

62 

81.66 

6.48 

61.63 

6.75 

61.60 

V.  02  61.57 

7.29 

62 

63 

62.65 

6.59 

62.63 

6.86 

62.60 

7.131  62.56 

7.40 

63 

64 

63.65 

6.69 

63.62 

6.97 

63.59 

7.25  163.56 

7.52 

64 

65 

64.64 

6.79 

64.61 

7.08 

64.58 

7.36 

64.55 

7.64 

65 

66 

65.64 

6.90 

65.61 

7.19 

65.58 

7.47 

65.54 

7.76 

66 

67 

66.63 

7.00 

66.60 

7.29 

66.57 

7  58 

66.54 

7.88 

67 

68 

67.63 

7.11 

67.60 

7.40 

67.56 

7.70 

67.53 

7.99 

68 

69 

68.62 

7.21 

68.59 

7.51 

68.56 

7.81 

68.52 

8.11 

69 

70  (69.62 

7.32 

69.58 

7.62 

69.55 

7.92 

69.51 

8.23 

70 

71.  70.61 

7.42 

70.58 

7.73 

70.54 

8.04  170.51 

8.35 

71 

72 

71.61 

7.53 

71.57 

7.84 

71.54 

8.15  71.50 

8.46 

72 

73 

72.60 

7.63 

72.57 

7.95 

72.53 

8.26  (72.49 

8.58 

73 

74 

73.59 

7.74 

73.56 

8.06 

73.52 

8.38 

'73.49 

8.70 

74 

75 

74.59 

7.84 

74.55 

8.17 

74.52 

8.49 

74.48 

8.82 

75 

76 

75.58 

7.94 

75.55 

8.27 

75.51 

8.60 

i75.47 

8.93 

76 

77 

7U.58 

8.05 

76.54 

8.38 

76.51 

8.72 

176.47 

9.05 

77 

78 

77.57 

8.15 

77.54 

8.49 

77.50 

8.83 

,77.46 

9.17 

78 

79 

78.57 

8.26 

78.53 

8.60 

78.49 

8.94 

i78.45 

9.29 

79 

80 

79.56 

8.36 

79.53 

8.71 

79.49 

9.06 

79.45 

9.40 

80 

81 

80.56 

8.47 

80.52 

8.82 

80.48 

~~9~.  17 

J80.44 

9.52 

81 

82 

81.55 

8.57 

81.51 

8.93 

81.47 

9!  28 

•81.43 

9.64 

82 

83 

82.55 

8.68 

82.51 

9.04 

82.47 

0.40 

'82.42 

9.76 

83 

84 

83.54 

8.78 

83.50 

9.14 

S3.  46 

9.51 

183.42 

9.87 

84 

85 

84.53 

8.88 

84.50 

9.25 

84.45 

9.62 

;  84.41 

9.99 

85 

86 

85.53 

8.99 

85.49 

9.36 

85.45 

9.74 

85.40 

10.11 

86 

87 

86.52 

9.09 

86.48 

9.47 

86.44 

9.85 

186.40 

10.23 

87 

88 

87.52 

9.20 

87.48 

9.58 

87.43 

9.96 

87.39 

10.34 

88 

89 

88.51 

9.30 

88.47 

9.69 

88.43 

10.08 

188.38 

10.46 

89 

90(89.51   9.41J 

89.47 

9.80 

89.42 

10.19 

189.38 

10.58 

90 

91  J90.50 

9.51  | 

90.46 

9.91 

90.42 

10.30 

190.37 

10.70 

91 

92  191.50 

9.62 

91.45 

10.02 

91.41 

10.41 

i91.36 

10.81 

92 

93 

92.49 

9.72 

92.45 

10.12 

92.40 

10.53 

i  92.36 

10.93 

93 

94 

93.49 

9.83 

93.44 

10.23 

93.40 

10.64 

93.35 

11.05 

94 

95 

94.48 

9.93 

94.44 

10.34 

94.39 

10.75 

94.34 

11.17 

95 

96 

95.47 

10.03 

95.43 

10.45 

95.38 

10.87 

95.33 

11.28 

96 

97  196.47 

10.14 

96.42 

10.56 

96.38 

10.98  !!96.33 

11.40 

97 

98  97.46 

10.24 

97.42 

10.67 

97.3V 

11.09 

97.32 

11.52 

98 

991  98.46 

10.35 

98.41 

10.78 

98.36 

11.21 

98.31 

11.64 

99 

100 

99.45 

10..  45 

99.41 

10.89 

99.36 

11.  33  ||  99.31 

11.75 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  1  Dep. 

Lat. 

1 

P 

84  Deg. 

83|  Deg. 

83|  Deg.     83i  Deg. 

rt 

3- 

16 


TRAVERSE    TABLE- 


o 

7  Deg. 

1\  Deg. 

7^Deg. 

71  Deg. 

C 

5' 

§ 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

§ 
P 

i 

0.99 

0.12 

0.99 

0.13 

0.99 

0.13 

0.99 

O.J3 

1 

2 

1.99 

0.24 

1.98 

0.25 

1.98 

0.26 

1.98 

0.27 

2 

3 

2.98 

0.37 

2.98 

0.38 

2.97 

0.39 

2.97 

0.40 

3 

4 

3.97 

0.49 

3.97 

0.50 

3.97 

0.52 

3.96 

0.54 

4 

5 

4.96 

0.61 

4.96 

0.63 

4.96 

0.65 

4.95 

0.67 

5 

6 

5.96 

0.73 

5.95 

0.76 

5.95 

0.78 

5.95 

0.81 

6 

7 

6.95 

0.85 

6.94 

0.88 

6.94 

0.91 

6.94 

0.94 

7 

8 

7.94 

0.97 

7.94 

1.01 

7.93 

1.04 

7.93 

.08 

8 

9 

8.93 

1.10 

8.93 

1.14 

8.92 

1.17 

8.92 

.21 

9 

10 

9.93 

1.22 

9.92 

1.26 

9.91 

1.31 

9.91 

.35 

10 

11 

10.92 

1.34 

10.91 

1.39 

10.91 

1.44 

10.90 

.48 

11 

12 

11.91 

.46 

11.90 

1.51 

11.90 

1.57 

11.89 

.62 

12 

13 

12.90 

.58 

12.90 

1.64 

12.89 

1.70 

12.88 

.75 

13 

14 

13.90 

.71 

13.89 

1.77 

13.88 

1.83 

13.87 

.89 

14 

15 

14.89 

.83 

14.88 

1.89 

14.87 

1.96 

14.86 

2.02 

15 

16 

15.88 

.95 

15.87 

2.02 

15.86 

2.09 

15.85 

2.16 

16 

17 

16.87 

2.07 

16.86 

2.15 

16.85 

2.22 

16.84 

2.29 

17 

18 

17.87 

2.19 

17.86 

2.27 

17.85 

2.35 

17.84 

2.43 

18 

19 

18.86 

2.32 

18.85 

2.40 

18.84 

2.48 

18.83 

2.56 

19 

20 

19.85 

2.44 

19.84 

2.52 

19.83 

2.61  I 

19.82 

2.70 

20 

21 

20.84 

2.56 

20.83 

2.65 

20.82 

2.74 

20.81 

2.83 

21 

22 

21.84 

2.68 

21.82 

2.78 

21.81 

2.87 

21.80 

2.97 

22 

23 

22.83 

2.80 

22.82 

2.90 

22.80 

3.00 

22.79 

3.10 

23 

24 

23.82 

2.92 

23.81 

3.03 

23.79 

3.13 

23.78 

3.24 

24 

25 

24.81 

3.05 

24.80 

3.15 

24.79 

3.26 

24.77 

3.37 

25 

26 

25.81 

3.17 

25.79 

3.28 

25.78 

3.39 

25.76 

3.51 

26 

27 

26.80 

3.29 

26.78 

3.41 

26.77 

3.52 

26.75 

3.64 

27 

28 

27.79 

3.41 

27.78 

3.53 

27.76 

3.65 

27.74 

3.78 

28 

29 

28.78 

3.53 

28  .  77 

3.66 

28.75 

3.79 

28.74 

3.91 

29 

30 

29.78 

3.66 

29.76 

3.79 

29.74 

3.92 

29.73 

4.05 

30 

31 

30.77 

3.78 

30.75 

3.91 

30.73 

4.05 

30.72 

4.18 

31 

32 

31.76 

3.90 

31.74 

4.04 

31.73 

4.18 

31.71 

4.32 

32 

33 

32.75 

4.02 

32.74 

4.16 

32.72 

4.31 

32.70 

4.45 

33 

34 

33.75 

4.14 

33.73 

4.29 

33.71 

4.44 

33.69 

4.58 

34 

35 

34.74 

4.27 

34.72 

4.42 

34.70 

4.57 

34.68 

4.72 

35 

36 

35.73 

4.39 

35.71 

4.54 

35.69 

4.70 

35.67 

4.85 

36 

37 

36.72 

4.51 

36.70 

4.67 

36.68 

4.83 

36.66 

4.99 

37 

38 

37.72 

4.63 

37.70 

4.80 

37.67 

4.96 

37.65 

5.12 

38 

39 

38.71 

4.75 

38.69 

4.92 

38.67 

5.09 

38.64 

5.26 

39 

40 

39.70 

4.87 

39.68 

5.05 

39.66 

5.22 

39.63 

5.39 

40 

41 

40.70 

5.00 

40.67 

5.17 

40.65 

5.35 

40.63 

5.53 

41 

42 

41.69 

5.12 

41.66 

5.30 

41.64 

5.48 

41.62 

5.66 

42 

43 

42.68 

5.24 

42.66 

5.43 

42.63 

5.61 

42.61 

5.80 

43 

44 

43.67 

5.36 

43.65 

5.55 

43.62 

5.74 

43.60 

5.93 

44 

45 

44.67 

5.48 

44.64 

5.68 

44.62 

5.87 

44.59 

6.0'   45 

46 

45.66 

5.61 

45.63 

5.81 

45.61 

6.00 

45  58   6.20 

46 

47 

46.65 

5.73 

46.62 

5.93 

46.60 

6.13 

46.57 

6.34 

47 

48 

47.64 

5.85 

47.62 

6.06 

47.59 

6.27 

47.56 

6.47 

48 

49 

48.63 

5.97 

48.61 

6.18 

4S.58 

6.40 

48.55 

6.61 

49 

50 

49.63 

6.09 

49.60 

6.31 

49.57 

6.53 

49.54 

6.74 

50 

8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

CJ 

c 

W 

"K 

3 

83  Deg. 

82|  Deg. 

82|  Deg. 

82i  Deg. 

3 

TRAVERSE    TABLE. 


17 


2 

7  Deg. 

7£  Deg. 

7*  Deg. 

a 
sr 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

•50.62 

6.22 

50.59 

6.44 

50.56 

6.66i 

50.53 

6.88 

51 

52 

51.61 

6.34 

51.58 

6.56 

51.56 

6.79 

51.53 

7.01 

52 

53 

52.60 

6.46 

52.58 

6.69 

52.55 

6.92 

52  .  52 

7.15 

53 

54 

53.60 

6.58 

53.57 

6.81 

53.54 

7.05 

53.51 

7.28 

54 

55 

54.59 

6.70 

54.56 

6.94 

54.53 

7.18 

54  .  50 

7.42 

55 

56    55.58 

6.82 

55.55 

7.07 

55.52 

7.31 

55.49 

7.55 

56 

57    56.58 

6.95 

56.54 

7.19 

56.51 

7.44 

56.48 

7.69 

57 

58    57.57 

7.07 

57.54 

7.32 

57.50 

7.57 

57.47 

7.82 

58 

59    53.56 

7.19 

58.53 

7.45 

58.50 

7.70 

58.46 

7.96 

59 

60    59.55 

7.31 

59.52 

7.57 

59.49 

7.83 

59.45 

8.09 

60 

61     60.551 

7.43 

60.51 

7.70 

60.48 

7.96 

69.44 

8.23 

"H 

62    61.54 

7.56 

61.50 

7.82 

61.47 

8.09 

61.43 

8.36 

62 

63 

62.53 

7.68 

62.50 

7.95 

62.46 

8.22 

62.42 

8.50 

63 

64 

63  .  52 

7.80 

63.49 

8.08 

63.45 

8.35 

63.42 

8.63 

64 

65 

64.52 

7.92 

64.48 

8.20 

64.44 

8.48 

64.41 

8.77 

65 

66 

65.51 

8.04 

65.47 

8.33 

65.44 

8.61 

65.40 

8.90 

66 

67 

66  50 

8.17 

66.46 

8.46 

66.43 

8.75 

66.39 

9.04 

67 

68 

67.49 

8.29 

67.46 

8.58 

67.42 

8.88 

67.38 

9.17 

68 

69 

68.49 

8.41 

68.45 

8.71 

68.41 

9.01 

68.37 

9.30 

69 

70 

69.48 

8.53 

69.44 

8.83 

69.40 

9.14 

69.36 

9.44 

70 

71 

70.47 

8.65 

70.43 

8.96 

70,30 

9.27 

70.35 

9.57 

71 

72 

71.46 

8.77 

71.42 

9.09 

71.38 

9.40 

71.34 

9.71 

72 

73 

72.46 

8.90 

72.42 

9.21 

72  38 

9.53 

72.33 

9.84 

73 

74 

73.45 

9.02 

73.41 

9.34 

73.37 

9.66 

73.32 

9.98 

74 

75 

74.44 

9.14 

74.40 

9.46 

74.36 

9.79 

74.31 

10.11 

75 

76 

75.43 

9.26 

75.39 

9.59 

75.35 

9.92 

75.31 

10.25 

76 

77 

76.43 

9.38 

76.38 

9.72 

76.34 

10.05 

76.30 

10.38 

77 

78 

77.42 

9.51 

77.38 

9.84 

77.33 

10.18 

77.29 

10.52 

78 

79 

78.41 

9.63 

78.37 

9.97 

78.32 

10.31 

78.28 

10.65 

79 

80 

79.40 

9.75 

79>.36 

10.10 

79.32 

10.44 

79.27 

10.79 

80 

81 

80.40 

9.87 

80.35 

10.22 

80.31 

10.57 

80.26 

10.92 

81 

82 

81.39 

9.99 

81.34 

10.35 

81.30 

10.70 

81.25 

11.06 

82 

83 

82.38 

10.12 

82.34 

10.47 

82.29 

10.83 

82.24 

11.19 

83 

84 

83.37 

10.24 

83.33 

10.60 

83.28 

10.96 

83.23 

11.33 

84 

85 

84.37 

10.36 

84.32 

10.73 

84.27 

11.09 

84.22 

11.46 

85 

86 

85.36 

10.48 

85.31 

10.85 

85.26 

11.23 

85.21 

11.60 

86 

87 

86.35 

10.60 

86.30 

10.98 

86.26 

11.36 

86.21 

11.73 

87 

88 

87.34 

10.72 

87.30 

11.11 

87.25 

11.49 

87.20 

11.87 

88 

89 

88.34 

10.85 

88.29 

11.23 

88.24 

11.62 

88.19 

12.00 

89 

90 

89.33 

10  97 

89.28 

11.36 

89.23 

11.75 

89.18 

12.14 

90 

91 

90.32 

11.09 

90.27 

11.48 

90.22 

11.88 

90U7 

12.27 

91 

92 

91.31 

11.21 

91.26 

11.61 

91.21 

12.  Oi 

91.16 

12.41 

92 

93 

92.31 

11.33 

92.26 

11.74 

92.20 

12.14 

92.15 

12.54 

93 

94 

93.30 

11.46 

93.25 

11.86 

93.20 

12.27 

93.14 

12.68 

94 

95 

94.29 

11.58 

94.24 

11.99 

94.19 

12.40 

94.13 

12.81 

95 

96    95.28 

11.70 

95.23 

12.12 

95.18 

12.53 

95.12 

12.95 

96 

97 

96.28 

11.82 

96.22 

12.24 

96.17 

12.66 

96.11 

13.08 

97 

98 

97.27 

11.94 

97,22 

12.37 

97.16 

12.79 

97.10 

13.22 

98 

99 

93.26 

12.07 

98.21 

12.49 

98.15 

12.92 

98.10 

13.35 

99 

100    99.25 

12.19 

99.20 

12.62 

99.14 

13.05 

99.09 

13.49 

100 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

~ 

.-i 

83  Deg. 

821  Deg. 

82ADeg. 

82J  Degr. 

Q 

18 


TRAVERSE    TABLE. 


o 

8  Deg. 

8*  Deg. 

8£  Deg. 

8|  Deg. 

3 

stance. 

stance.! 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.99 

0.14 

0.99 

0.14| 

0.99 

0.15 

0.99 

0.15 

1 

2 

1.98 

0.28 

1.98 

0.29 

1.98 

0.30 

1.98 

0.30 

2 

3 

2.97 

0.42 

2.97 

0.43 

2.97 

0.44 

2.97 

0.46 

3 

4 

3.96 

0.56 

3.96 

0.57 

3.96 

0.59 

3.95 

0.61 

4 

5 

4.95 

0.70 

4.95 

0.72 

4.95 

0.74 

4.94 

0.76 

5 

6 

5.94 

0.84 

5.94 

0.86 

5.93 

0.89 

5.93 

0.91 

6 

7 

6.93 

0.97 

6.93 

1.00 

6.92 

1.03 

6.92 

1.06 

7 

8 

7.92 

1.11 

7.92 

1.15 

7.91 

1.18 

7.91 

1.22 

8 

9 

8.91 

1.25 

8.91 

1.29 

8.90 

1.33 

8.90 

1.37 

9 

10 

9.90 

1.39 

9.90 

1.43 

9.89 

1.48 

9.88 

1.52 

10 

11 

10.89 

1.53 

;0.89 

1.58 

10.88 

1.63 

10.87 

"l.67 

11 

12 

11.88 

1.67 

11.88 

1.72 

11.87 

1.77 

11.86 

1.83 

12 

13 

12.87 

1.81 

12.87 

1.87 

12.86 

1.92 

12.85 

1.98 

13 

14 

13.86 

1.95 

13.86 

2.01 

13.85 

2.07 

13.84 

2.13 

14 

15 

14.85 

2.09 

14.85 

2.15 

14.84 

2.22 

14.83 

2.28 

15 

16 

15.84 

2.23 

15.84 

2.30 

15.82 

2.36 

15.81 

2.43 

16 

17 

16.83 

2.37 

16.83 

2.44 

16.81 

2.51 

16.80 

2.59 

17 

IS 

17.82 

2.51 

17.81 

2.58 

17.80 

2.66 

17.79 

2.74 

18 

19 

18.82 

2.64 

18.80 

2.73 

18.79 

2.81 

18.78 

2.89 

19 

20 

19.81 

2.78 

19.79 

2.87 

19.78 

2.96 

19.77 

3.04 

20 

21 

20.80 

2.92 

20.78 

3.01 

20.77 

3.10 

20.76 

3.19 

21 

22 

21.79 

3.06 

21.77 

3.16 

21.76 

3.25 

21.74 

3.35 

22 

23 

22.78 

3.20 

22.76 

3.30 

22.75 

3.40 

22.73 

3.50 

23 

.24 

23.77 

3.34 

23.75 

3.44 

23.74 

3.55 

23.72 

3.65 

24 

25 

24.76 

3.48 

24.74 

3.59 

24.73 

3.70 

24.71 

3.80 

25 

26 

25.75 

3.62 

25.73 

3.73 

25.71 

3.84 

25.70 

3.96 

26 

27 

26.74 

3.76 

26.72 

3.87 

26.70 

3.99 

26.69 

4.11 

27 

28 

27.73 

3.90 

27.71 

4.02 

27.69 

4.14 

27.67 

4.26 

28 

29 

28.72 

4.04 

28.70 

4.16 

28.68 

4.29 

28.66 

4.41 

29 

30 

2,9.71 

4.18 

29.69 

4.30 

29.67 

4.43 

29.65 

4.56 

30 

31 

30.70 

4.31  | 

30.68 

4.45 

30.66 

4.58 

30.64 

4.72 

31 

32 

31.69 

4.45 

31.67 

4.59 

31.65 

4-.  73 

31.63 

4.87 

32 

33 

32.68 

4.59 

32.66 

4.74 

32.64 

4.88 

32  .  62 

5.02 

33 

34 

33.67 

4.73 

33.65 

4.88 

33.63 

5.03 

33.60 

5.17 

34 

35 

34.66 

4.87 

34.64 

5.02 

34.62 

5.17 

34.59 

5.32 

35 

36 

35.65 

5.01 

35.63 

5.17 

35.60 

5.32 

35.58 

5.48 

36 

37 

36.64 

5.15 

36.62 

5.31 

36.59 

5:47 

36.57 

5.63 

37 

38 

37.63 

5.29 

37.61 

5.45 

37.58 

5.62 

37.56 

5.78 

38 

39 

38.62 

5.43 

38.60 

5.60 

38.57 

5:76 

38.55 

5.93 

39 

40 

39.61 

5.57 

39  .  59 

5.74 

39.56 

5.91 

39.53 

6.08 

40 

"41 

40.60 

5.71 

40.58 

5.88 

40  .  55 

6.06 

40.52 

6.24 

41 

42 

41.59 

5.85 

41.57 

6.03 

41.54 

6.21 

41.51 

6.39 

42 

43 

42.58 

5.98 

42.56 

6.17 

42.53 

6.36 

42.50 

6.54 

43 

44 

43.57 

6.12 

43.54 

6.31 

43.52 

6.50 

43.49 

6.69 

44 

45 

44.56 

6.26 

44.53 

6.46 

44.51 

6.65 

44.48 

6.85 

45 

46 

45.55 

6.40 

45.52 

6.60 

45.49 

6.80 

45.46 

7.00 

46 

47 

46.54 

6.54 

46.51 

6.74 

46.48 

6.95 

46.45 

7.15 

47 

•  48 

47.53 

6.68 

47.50 

6.89 

47.47 

7.09 

47.44 

7.30 

48 

49 

48.52 

6.82 

48.49 

7.03 

48.46 

7.24 

48.43 

7.45 

49 

50 

49.51 

6.96 

49.48 

7.17 

49.45 

7.39 

49.42 

7.61 

50 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

s 

82  Deg. 

81J  Deg.  - 

8t£  Deg. 

81i  Deg. 

P 

TRAVERSE    TABLE. 


19 


c 

8  Deg. 

»i  Deg. 

8*  Deg. 

S|  Deg. 

g 

CO 

1 

1' 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

1 

51 

50.50 

7.10 

50.47 

7.32 

50.44 

7.54 

50.41 

7.76 

51 

52 

51.49 

7.24 

51.46 

7.46 

51.43 

7.69 

51.39 

7.91 

52 

53 

52.48 

7.38 

52.45 

7.61 

52.42 

7.83 

52.38 

8.06 

53 

54 

53.47 

7.52 

53.44 

7.75 

53.41 

7.98 

53.37 

8.21 

54 

55 

54.46 

7.65 

54.43 

7.89 

54.40 

8.13 

54.36 

8.37 

55 

56 

55.46 

7.79 

55.42 

8.04 

55.38 

8.28 

55.35 

8.52 

56 

57 

56.45 

7.93 

56.41 

8.18 

56.37 

8.43 

56.34 

8.67 

57 

58 

57.44 

8.07 

57.40 

8.32 

57.36 

8.57 

57.32 

8.82 

58 

59 

58.43 

8.21 

58.39 

8.47 

58.35 

8.72 

58.31 

8.98 

59 

60 

59.42 

8.35 

59.38 

8.61 

59.34 

8.87 

59.30 

9.13 

60 

61 

60.41 

8.49 

60.37 

8.75 

60.33 

9.02 

60.29 

"  9.28 

61 

62 

61.40 

8.63 

61.36 

8.90 

61.32 

9.16 

61.28 

9.43 

62 

63 

62.39 

8.77 

62.35 

9.04 

62.31 

r9.31 

62.27 

9.58 

63 

64 

63.38 

8.91 

63.34 

9.18 

63.30 

9.46 

63.26 

9.74 

64 

65 

64.37 

9.05 

64.33 

9.33 

64.29 

9.61 

64.24 

9.89 

65 

66 

65.36 

9.19 

65.32 

9.47 

65.28 

9.76 

65.23 

10.04 

66 

67 

66.35 

9.32 

66.31 

9.61 

66.26 

9.90 

66.22 

10.19 

67 

68 

67.34 

9.46 

67.30 

9.76 

67.25 

10.05 

67.21 

10.34 

68 

69 

68.33 

9.60 

68.29 

9.90 

68.24 

10.20 

68.20 

10.50 

69 

•70 

69.32 

9.74 

69.28 

10.04 

69.23 

10.35 

69.19 

10.65 

70 

71 

70.31 

9.88. 

70.27 

10.19 

70.22 

10.49 

70.17 

10.80 

71 

72 

71.30 

10.02 

71.25 

10.33 

71.21 

10.64 

71.16 

10.95 

72 

73 

72.29 

10.16 

72.24 

10.47 

72.20 

10.79 

72.15 

11.10 

73 

74 

73.28 

10.30 

73.23 

10.62 

73.19 

10.94 

73.14 

11.26 

74 

75 

74.27 

10.44 

74.23 

10.76 

74.18 

11.09 

74.13 

11.41 

75 

78 

75.26 

10.58 

75.21 

10.91 

75.17 

11.23 

75.12 

11.56 

76 

77 

76.25 

10.72 

76.20 

11.05 

76.15 

11.38 

76.10 

11.71 

77 

78 

77.24 

10.86 

77.19 

11.19 

77.14 

11.53 

77.09 

11.87 

78 

79 

78.23 

10.99 

78.18 

11.34 

78.13 

11.68 

73.08 

12.02 

79 

80 

79.22 

11.13 

79.17 

11.48 

79.12 

11.82 

79.07 

12.17 

80 

81 

80.21 

11.27 

80.16 

11.62 

80.11 

11.97 

80.06 

12.32 

81 

82 

81.20 

11.41 

81.15 

11.77 

81.10 

12.12 

81.05 

12.47 

82 

83 

82.19 

11.55 

82.14 

11.91 

82.09 

12.27 

82.03 

12.63 

83 

84 

83.18 

11.69 

83.13 

12.05 

83.08 

12.42 

83.02 

12.78 

84 

85 

84.17 

11.83 

84.12 

12.20 

84.07 

12.56 

84.01 

12.93 

85 

86 

85.16 

11.97 

85.11 

12.34 

85.06 

12.71 

85.00 

13.08 

86 

87 

86.15 

12.11 

86.10 

12.48 

86.04 

12.86 

85.99 

13.23 

87 

88 

87.14 

12.25 

87.09 

12.63 

87.03 

13.01 

86.98 

13.39 

88 

89 

88.13 

12.39 

88.08 

12.77 

88.02 

13.16 

87.96 

13.54 

89 

90 

89.12 

12.53 

89.07 

12.91 

89.01 

13.  £0 

88.95 

13.69 

90 

31 

90.11 

12.66 

90.06 

13.06 

90.00 

13.45 

89.94 

13.84 

91 

92 

91.10 

12.80 

91.05 

13.20 

90.99 

13.60 

90.93 

14.00 

92 

93 

92.09 

12.94 

92.04 

13.34 

91.98 

13.75 

91.92 

14.15 

93 

94 

93.09 

13.08 

93.03 

13.49 

92.97 

13.89 

92.91 

14.30 

94 

95 

94.08 

13.22 

94.02 

13.63 

93.96 

14.04 

93.89 

14.45 

95 

96 

95.07 

13.36 

95.01 

13.78 

94.95 

14.19 

94.88 

14.60 

96 

97 

96.06 

13.50 

96.00 

13.92 

95.93 

14.34 

95.87 

14.76 

97 

98 

97.05 

13.64 

96.99 

14.06 

96.92 

14.49 

96.86 

14.91 

98 

99 

98.04 

13.78 

97.98 

14.21 

97.91 

14.63 

97.85 

15.06 

99 

100 

99.03 

13.92 

98.97 

14.35 

98.90 

14.78 

98.84 

15.21 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

1 
Q 

82  Deg. 

C1J  Deg. 

Sl^Deg. 

814  Deg. 

J 
.2 

Q 

TRAVERSE    TABLE. 


g 

9  Deg. 

9}  Deg. 

9i  Deg. 

9|  Deg. 

C 

1 

CD 

stance.l 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.99 

0.16 

0.99 

0.16 

0.99 

0.17 

0.99 

0.17 

1 

2 

1.98 

0.31 

1.97 

0.32 

1.97 

0.33 

1.97 

0.34 

2 

3 

2.96 

0.47 

2.96 

0.48 

2.96 

0.50 

2.96 

0.51 

3 

4 

3.95 

0.63 

3.95 

0.64 

3.95 

0.66 

3.94 

0.68 

4 

5 

4.94 

0.78 

4.93 

0.80 

4.93 

0.83 

4.93 

0.85 

5 

6 

5.93 

0.94 

5.92 

0.96 

5.92 

0.99 

5.91 

1.02 

6 

7 

6.91 

1.10 

6.91 

1.13 

6.90 

1.16 

6.90 

1.19 

7 

8 

7.90 

1.25 

7.90 

1.29 

7.89 

1.32 

7.88 

1.35 

8 

9 

8.89 

1.41 

8.88 

1.45 

8.88 

1.49 

8.87 

1.52 

9 

10 

9.88 

1.56 

9.87 

1.61 

,  9.86 

1.65 

9.86 

1.69 

10 

11 

10.86 

1.72 

10.86 

1.77 

10.85 

1.82 

10.84 

1.86 

11 

12 

11.85 

1.88 

11.84 

1.93 

11.84 

1.98 

11.83 

2,03 

12 

13 

12.84 

2.03 

12.83 

2.09 

12.82 

2.15 

12.81 

2.20 

13 

14 

13.83 

2.19 

13.82 

2.25 

13.81 

2.31 

13.80 

2.37 

14 

15 

14.82 

2.35 

14.80 

2.41 

14.79 

2.48 

14.78 

2.54 

15 

16 

15.80 

2.50 

15.79 

2.57 

15.78 

2.64 

15.77 

2.71 

16 

17 

16.79 

2.66 

16.78 

2.73 

16.77 

2.81 

16.75 

2.88 

17 

18 

17.78 

2.82 

17.77 

2.89 

17.75 

2.97 

17.74 

3.05 

13 

19 

18.77 

2.97 

18.75 

3.05 

18.74 

3.14 

IS.  73 

3.22 

19 

20 

19.75 

3.13 

19.74 

3.21 

19.73 

3.30  j 

19.71 

3.39 

20' 

21 

20.74 

3.29 

20.73 

3.38 

20.71 

3.47 

20.70 

3.56 

21 

22 

21.73 

3.44 

21.71 

3.54 

21.70 

3.63 

21.68 

3.73 

22 

23 

22.72 

3.60 

22.70 

3.70 

22.68 

3.80 

22.67 

3.90 

23 

24 

23.70 

3.75 

23.69 

3.86 

23.67 

3.96 

23.65 

4.06 

24 

25 

24.69 

3.91 

24.67 

4.02 

24.66 

4.13 

24.64 

4.23 

25 

26 

25.68 

4.07 

25.66 

4.18 

25.64 

4.29 

25.62 

4.40 

26 

27 

26.67 

4.22 

26.65 

4.34 

26.63 

4.46 

26.61 

4.57 

27 

28 

27.66 

4.38 

27.64 

4.50 

27.62 

4.62 

27.60 

4.74 

28 

29 

28  ,  64 

4.54 

28.62 

4.66 

28.60 

4.79 

28.58" 

4.91 

29 

30 

29.63 

4.69 

29.61 

4.82 

29.59 

4.95 

29.57 

5.08 

30 

31 

30.62 

4.85 

30.60 

4.98 

30.57 

5.12 

30.55 

5.25 

31 

32 

31.61 

5.01 

31.58 

5.14 

31.56 

5.28 

31.54 

5.42 

32 

33 

32.59 

5.16 

32.57 

5.30 

32.55 

5.45 

32.52 

5.59 

33 

34 

33.58 

5.32 

33.56 

5.47 

33.53 

5.61 

33.51 

5.76 

34 

35 

34.57 

5.48 

34.54 

5.63 

34.52 

5.78 

34.49 

5.93 

35 

36 

35.56 

5.63 

35.53 

5.79 

35.51 

5.94 

35.48 

6.10 

36 

37 

36.54 

5.79 

36.52 

5.95 

36.49 

6.11 

36.47 

6.27 

37 

38 

37.53 

5.94 

37.51 

6.11 

37.48 

6.27 

37.45 

6.44 

38 

39 

38.52 

6.10 

38.49 

6.27 

38.47 

6.44 

38.44 

6.60 

39 

40 

39.51 

6.26 

39.48 

6.43 

39.45 

6.60 

39.42 

6.77 

40 

11 

40.50 

6.41 

40.47 

6.59 

40.44 

6.77 

40.41 

6.94 

41 

42 

41.48 

6.57 

41.45 

6.75 

41.42 

6.92 

41.39 

7.11 

42 

43 

42.47 

6.73 

42.44 

6.91 

42.41 

7.10 

42.38 

7.28 

43 

44 

43.46 

6.88 

43.43 

7.07 

43.40 

7.26 

43.36 

7,45 

44 

45 

44.45 

7.04 

44.41 

7.23 

44.38 

7.43 

44.35 

7.62 

45 

46 

45.43 

7.20 

45.40 

7.39 

45.37 

7.59 

45.34 

7.79 

46 

47 

46.42 

7.35 

46.39 

7.55 

46.36 

7.76 

46.32 

7.96 

47 

48 

47.41 

7.51 

47.38 

7.72 

47.34 

7.92 

47.31 

8.13 

48 

49 

48.40 

7.67 

48.36 

7.88 

48.33 

8.09 

48.29 

8.30 

49 

50 

49.38 

7.82 

49.35 

8.04 

49.32 

8.25 

49.28 

8.47 

50 

«' 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

.2 
Q 

81  Deg. 

80|  Deg. 

801  Deg. 

80i  Deg. 

1 

3 

TRAVERSE    TABLE. 


1 

9 

5 

6 

9  Deg. 

94  Deg. 

9^  Deg. 

9|  Deg. 

q 

P 

? 

I 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

51 

50.37 

7.98 

50.34 

8.20 

50.30 

8.42 

50.26 

8.64 

51 

52 

51.36 

8.13 

51.32 

8.36 

51.29 

8.58 

51.25 

8.81 

52 

53 

52.35 

8.29 

52.31 

8.52 

52.27 

8.75 

52.23 

8.98 

53 

54 

53.34 

8.45 

53.30 

8.68 

53.26 

8.91 

53.22 

9.14 

54 

55 

54.32 

8.60 

54.28 

8.84 

54.25 

9.08 

54.21 

9.31 

55 

56 

55.31 

8.76 

55.27 

9.00 

55.23 

9.24 

55.19 

9.48 

56 

57 

56.30 

8.92 

56.26 

9.16 

56.22 

9.41 

56.18 

9.65 

57 

58 

57.29 

9.07 

57.25 

9.32 

57.20 

9.57 

57.16 

9.82 

58 

59 

58.27 

9.23 

58.23 

9.48 

58.19 

9.74 

58.15 

9.99 

59 

60 

59.26 

9.39 

59.22 

9.64 

59.18 

9.90 

59.13 

10.16 

60 

61 

60.25 

9.54 

60.21 

9.81 

60.16 

10.07 

60.12 

10.33 

6] 

62 

61.24 

9.70 

61.19 

9.97 

61.15 

10.23 

61.10 

10.50 

62 

63 

62.22 

9.86 

62.18 

10.13 

62.14 

10.40 

62.09 

10.67 

63 

64 

63.21 

10.01 

63.17 

10.29 

63.12 

10.56 

63.08 

10.84 

64 

65 

64.20 

10.17 

64.15 

10.40 

64.11 

10.73 

64.06 

11.01 

65 

66 

65.19 

10.32 

65.14 

10.61 

65.09 

10.89 

65.05 

11.18 

66 

67 

66.18 

10.48 

66.13 

10.77 

66.08 

11.06 

66.03 

11.35 

67 

68 

67.16 

10.64 

67.12 

10.93 

67.07 

11.22 

67.02 

11.52 

68 

69 

68.15 

10.79 

68.10 

11.09 

68.05 

11.39 

68.00 

11.69 

69 

70 

69.14 

10.95 

69.09 

11.25 

69.04 

11.55 

68.99 

11.85 

70 

71 

70.13 

11.11 

70.08 

11.41 

70.03 

11.72 

69.97 

12.02 

71 

72 

71.11 

11.26 

71.06 

11.57 

71.01 

11.88 

70.96 

12.19 

72 

73 

72.10 

11.42 

72.05 

11.73 

72.00 

12.05 

71.95 

12.36 

73 

74 

73.09 

11.58 

73.04 

11.89 

72.99 

12.21 

72.93 

12.53 

74 

75 

74.08 

11.73 

74.02 

12.06 

73.97 

12.38 

73.92 

12.70 

75 

76 

75.06 

11.89 

75.01 

12.22 

74.96 

12.54 

74.90 

12.87 

76 

77 

76.05 

12.05 

76.00 

12.38 

75.94 

12.71 

75.89 

13.04 

77 

78 

77.04 

12.20 

76.99 

12.54 

76.93 

12.87 

76.87 

13.21 

78 

79 

78.03 

12.36 

77.97 

12.70 

77.92 

13.04 

77.86 

13.38 

79 

80 

79.02 

12.51 

78.96 

12.86 

78.90 

13.20 

78.84 

13.55 

80 

81 

80.00 

12.67 

79.95 

13.02 

79.89 

13.37 

79.83 

13.72 

81 

82 

80.99 

12.83 

80.93 

13.18 

80.88 

13.53 

80.82 

13.89 

82 

83 

81.98 

12.98 

81.92 

13.34 

81.86 

13.70 

81.80 

14.06 

83 

84 

82.97 

13.14 

82.91 

13.50 

82.85 

13.86 

82.79 

14.23 

84 

85 

83.95 

13.30 

83.89 

13.66 

83.83 

14.03 

83.77 

14.39 

85 

86 

84.94 

13.45 

84.88 

13.82 

84.82 

14.19 

84.76 

14.56 

86 

87 

85.93 

13.61 

85.87 

13.98 

85.81 

14.36 

85.74 

14.73 

87 

88 

86.92 

13.77 

86.86 

14.15 

86.79 

14.52 

86.73 

14.90 

88 

89 

87.90 

13.92 

87.84 

14.31 

87.78 

14.69 

87.71 

15.07 

89 

90 

88.89 

14.08 

88.83 

14.47 

88.77 

14.85 

88.70 

15.24 

90 

91 

89.88 

14.24 

89.82 

14.63 

89.75 

15.02 

89.69 

15.41 

91 

92 

90.87 

14.39 

90.80 

14.79 

90.74 

15.18 

90.67 

15.58 

92 

93 

91.86 

14.55 

91.79 

14.95 

91.72 

15.35 

91.66 

15.75 

93 

94 

92.84 

14.70 

92.78 

15.11 

92.71 

15.51 

92.64 

15.92 

94 

95 

93.83 

14.86 

93.76 

15.27 

93.70 

15.68 

93.63 

16.09 

95 

96 

94.82 

15.02 

94.75 

15.43 

94.68 

15.84 

94.61 

16.26 

96 

97 

95.81 

15.17 

95.74 

15.59 

95.67 

16.01 

95.60 

16.43 

97 

98 

96.79 

15.33 

96.73 

15.75 

96.66 

16.17 

96.58 

16.60 

98 

99 

97.78 

15.49 

97.71 

15.91 

97.64 

16.34 

97.57 

16.77 

99 

100 

98.77 

15.64 

98.70 

16.07 

98.63 

16.50 

98.56 

16.93 

100 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

Q 

81  Deg. 

80|  Deg. 

80f  Deg. 

804  Deg. 

-"; 

s 

.   *.  t. 

*•#*" 


TEA VERSE    TABLE. 


o 

10  Deg. 

10i  Deg. 

10|  DeS- 

10|  Deg. 

g 

r3 

D 

o 

o 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 
E 

o 

CO 

1 

0.98 

0.17 

0.98 

0.18 

0.98 

0.18 

0.98 

0.19 

1 

2 

1.97 

0.35 

1.97 

0.36 

1.97 

0.36 

1.96 

0.37 

3 

2.95 

0.52 

2.95 

0.53 

2.95 

0.55 

2.95 

0.56 

3 

4 

3.94 

0.69 

3.94 

0.71 

3.93 

0.73 

3.93 

0.75 

4 

5 

4.92 

0.87 

4.92 

0.89 

4.92 

0.91 

4.91 

0.93 

5 

6 

5.91 

1.04 

5.90 

1.07 

5.90 

1.09 

5.89 

1  12 

6 

7 

6.89 

1.22 

6.89 

1.25 

6.88 

1.28 

6.88 

1.31 

7 

8 

7.88 

1.39 

7.87 

1.42 

7.87 

1.46 

7.86 

1.49 

8 

9 

8.86 

1.56 

8.86 

1.60 

8.85 

1.64 

8.84 

1.68 

9 

10 

9.85 

1.74 

9.84 

1.78 

9.83 

1.82 

9.82 

1.87 

10 

11 

10.83 

1.91 

10.82 

1.96 

10.82 

2.00 

10.81 

2.05 

11 

12 

11.82 

2.08 

11.81 

2.14 

11.80 

2.19 

11.79 

2.24 

12 

13 

12.80 

2.26 

12.79 

2.31 

12.78 

2.37 

12.77 

2.42 

13 

14 

13.79 

2.43 

13.78 

2.49 

13.77 

2.55 

13.75 

2.61 

14 

15 

14.77 

2.60 

14.76 

2.67 

14.75 

2.73 

14.74 

2.80 

15 

16 

15.76 

2.78 

15.74 

2.85 

15.73 

2.92 

15.72 

2.98 

16 

17 

16.74 

2.95 

16.73 

3.03 

16.72 

3.10 

16.70 

3.J7 

17 

18 

17.73 

3.13 

17.71 

3.20 

17.70 

3.28 

17.68 

3.36 

18 

19 

18.71 

3.30 

18.70 

3.38 

18.68 

3.46 

18.67 

3.54 

19 

20 

19.70 

3.47 

19.68 

3.56 

19.67 

3.64 

19.65 

3.73 

20 

21 

20.68 

3.65 

20.66 

3.74 

20.65 

3.83" 

20.63 

3.92 

21 

22 

21.67 

3.82 

21.65 

3.91 

21.63 

4.01 

21.61 

4.10 

22 

23 

22.65 

3.99 

22.63 

4.09 

22.61 

4.19 

22.60 

4.29 

23 

24 

23.64 

4.17 

23  .  62 

4.27 

23.60 

4.37 

23  .  58 

4.48 

24 

25 

24.62 

4-34 

24.60 

4.45 

24.58 

4.56 

24.56 

4.66 

25 

26 

25.61 

4.51 

25.59 

4.63 

25.56 

4.74 

25.54 

4.85 

26 

27 

26.59 

4.69 

26.57 

4.80 

26.55 

4.92 

26  .  53 

5.04 

27 

28 

27.57 

4.86 

27.55 

4.98 

27.53 

5.10 

27.51 

5.22 

28 

29 

28.56 

5.04 

28.54 

5.16 

28.51 

5.28 

28.49 

5.41 

29 

30 

29.54 

5.21 

29.52 

5.34 

29.50 

5.47 

29.47 

5.60 

30 

31 

30.53 

5.38 

30.51 

5.52 

30.48 

5.65 

30.46 

5.78 

31 

:  32 

31.51 

5.56 

31.49 

5.69 

31.46 

5.83 

31.44 

5.97 

32 

33 

32.50 

5.73 

32.47 

5.87 

32.45 

6.01 

32.42 

6.16 

33 

34 

33.48 

5.90 

33.46 

6.05 

33.43 

6.20 

33.40 

6.34 

34 

35 

34.47 

6.08 

34.44 

6.23 

34.41 

6.38 

34.39 

6.53 

35 

36 

35.45 

6.25 

35.43 

6.41 

35.40 

6.56 

35.37 

6.71 

35 

37 

36.44 

6.42 

36.41 

6.58 

36.38 

6.74 

36.35 

6.90 

37 

38 

37.42 

6.60 

37.39 

6.76 

37.36 

6.92 

37.33 

7.09 

38 

39 

38.41 

6.77 

38.38 

6.94 

38.35 

7.11 

38.32 

7.27 

39 

40 

39.39 

6.95 

39.36 

7.12 

39.33 

7.29 

39.30 

7.46 

40 

41 

40.38 

7.12 

40.35 

7.30 

40.31 

~7.47 

40.28 

7.65 

41 

42 

41.36 

7.29 

41.33 

7.47 

41.30 

7.65 

41.26 

7.83 

42 

43 

42.35 

7.47 

42.31 

7.65 

42.28 

7.84 

42.25 

8.02 

43 

44 

43.33 

7.64 

43.30 

7.83 

43.26 

8.02 

43.23 

8.21 

44 

45 

44.32 

7.81 

44.28 

8.01 

44.25 

8.20 

44.21 

8.39 

45 

46 

45.30 

7.69 

45.27 

8.19 

45.23 

8.38 

45.19 

8.58 

46 

47 

46.29 

8.16 

46.25 

8.36 

46.21 

8.57 

46.18 

8.77 

47 

48 

47.27 

8.34 

47.23 

8.54 

47.20 

8.75 

47.16 

8.95 

48 

49 

48.26 

8.51 

48.22 

8.72 

48.18 

8.93 

48.14 

9.14 

49 

50 

49.24 

8.68 

49.20 

8.90 

49.16 

9.11 

49.12 

9.33 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

c 

s 

80  Deg. 

79|  Deg. 

79|  Deg. 

79}  Deg. 

s 

TRAVERSE    TABLE. 


23 


p 

1' 

10  Deg. 

10J  Deg. 

101  Deg.     | 

10|  Deg. 

O 
S" 

2 
O 
Q 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

~5l 

50.23 

8.86 

50.19 

9.08 

50.15 

9  .-29 

50.10 

9.51 

51 

52 

51.21 

9.03 

51.17 

9.25 

51.13 

9.48 

51.09 

9.70 

52 

53 

52.19 

9.20 

52.15 

9.43 

52  .  1  1 

9.66 

52.07 

9.89 

53 

54 

53.18 

9.38 

53.14 

9.61 

53.10 

9.84 

53.05 

10.07 

54 

55 

54.16 

9.55 

54.12 

9.79 

54.08 

10.02 

04.03 

10.26 

55 

56 

55.15 

9.72J 

55.11 

9.96 

55.06 

10.21 

55.02 

10.45 

56 

57 

56.13 

9.90, 

56.09 

10.14 

56.05 

10.39 

56.00 

10.63 

57 

58    57.12 

10.07 

57.07 

10.32 

57.03 

10.57 

56.98 

10.82 

58 

59 

58.10 

10.25 

58.06 

10.50 

58.01 

10.75 

57.96 

11.00 

59 

60 

59.09 

10.42 

59.04 

10.68 

59.00 

10.93 

58.95 

11.19 

60 

61  ^60.07 

10.59 

60.Q3 

10.85 

59.98 

11.12 

59.93 

11.38 

61 

62 

61.06 

10.77 

61.01 

11.03 

60.96 

11.30 

60.91 

11.56 

62 

63 

62.04 

10.94 

61.99 

11.21 

61.95 

11.48 

61.89 

1*.75 

63 

64 

63.03 

11.11 

62.98 

11.39 

62.93 

11.66 

62.88 

11.94 

64 

65 

64.01 

11.29 

63.96 

11.57 

63.91 

11.85 

63.86 

12.12 

65 

66 

65.00 

11.46 

64.95 

11.74 

64.89 

12.03 

64.84 

12.31 

66 

67 

65.98 

11.63 

65.93 

11.92 

65.88 

12.21 

65.82 

12.50 

67 

68 

66.97 

11.81 

66.91 

12.10 

66.88 

12.39 

66.81 

12.68 

68 

69 

67.95 

11.98 

67.90 

12.28 

67.84 

12.57 

67.79 

12.  S7 

69 

70 

88.94 

12.16 

68.88 

12.46 

68.83 

12.76 

68.77 

13.06 

70 

71 

69.92 

12.33 

69.87 

12.63 

69.81 

12.94 

69.75 

13.24 

7] 

72. 

70.91 

12.50 

70.85 

12.81 

70.79 

13.12 

70.74 

13.43 

72 

73 

71.89 

12.68 

71.83 

12.99 

71.78 

13.30 

71.72 

13.62 

73 

74 

72.88 

12.85 

72.82 

13.17 

72.76 

13.49 

72.70 

13.80 

74 

75 

73.86 

13.02 

73.80 

13.35 

73.74 

13.67 

73.68 

13.99 

75 

76 

74.85 

13.20 

74.79 

13.52 

74.73 

13.85 

74.67 

14.18 

76 

77 

75.83 

13.37 

75.77 

13.70 

75.71 

14.03 

75.65 

14.36 

77 

78 

76.82 

13.54 

76.76 

13.88 

76.69 

14.21 

76.63 

14.55 

78 

79 

77.80 

13.72 

77.74 

14.06 

77.68 

14.40 

77.61 

14.74 

79 

80 

78.78 

13.89 

78.72 

14.24 

78.66 

14.58 

78.60 

14.92 

80 

81 

79.77 

14.07 

79.71 

14.41 

79.64 

14.76 

79.58 

15.11 

81 

82 

80.75 

14.24 

80.69 

14.59 

80.63 

14.94 

80.56 

15.29 

82 

83 

81.74 

14.41 

81.68 

14.77 

81.61 

15.13 

81.54 

15.48 

83 

84 

82.72 

14.59 

82.66 

14.95 

82.59 

15.31 

82.53 

15.67 

84 

85 

83.71 

14.76 

83.64 

15.13 

83.58 

15.49 

83.51 

15.85 

85 

86 

84.69 

14.93 

84.63 

15.30 

84.56 

15.67 

84.49 

16.04 

86 

87 

85.68 

15.11 

85.61 

15.48 

85.54 

15.85 

85.47 

16.23 

87 

88 

86.66 

15.28 

86.60 

15.66 

86.53 

16.04 

85.46 

16.41 

88 

89 

87.65 

15.45 

87.58 

15.84 

87.51 

16.22 

87.44 

16.60 

89 

90 

88.63 

15.63 

88.56 

16.01 

88.49 

16.40 

88.42 

16.79 

90 

91 

89.62    15.80 

89.55    16.19 

89.48 

16.53 

89.40 

16.97 

91 

92 

90.60    15.98 

90.53    16.37 

90.46 

16.77 

90.39 

17-16 

92 

93 

91.59 

16.15 

91.52    16.55 

91.44 

16.95 

91.37 

17.35 

93 

94 

92.57 

16.32 

92.50    16.73 

92.43 

17.13 

92.35 

17.53 

94 

95 

93.56 

16.50 

93.48    16.90 

93.41 

17.31 

93.33 

17.72 

95 

96 

94.54 

16.67 

94.47    17.08 

94.39 

17.49 

94.32 

17.91 

96 

97 

95.53 

16.84 

95.45    17.26 

95.38 

17.68 

95.30 

18.09 

97 

98 

96.51 

17.02 

96.44    17.44 

96.36 

17.86 

96.28 

18.28  1    98 

99 

97.50 

17.19 

97.42 

17.62 

97.34 

18.04 

97.26 

18.47 

99 

100 

98.48 

17.36 

98.40 

17.79 

98.33 

18.22 

98.25 

18.65 

100 

d 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

I 

80  Deg. 

79J  Deg. 

791  Deg. 

79i  Deg. 

CD- 

s 

TRAVERSE    TABLE. 


o 

5' 

11  Deg. 

Hi  Deg. 

'  11£  Deg. 

Ill  Deg. 

O 

«° 

? 

1 

& 
P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

p 

1 

0.98 

0.19 

0.98 

0.20 

0.98 

0.20 

0.98 

0.20 

1 

2 

1.96 

0.38 

1.96 

0.39 

1.96 

0.40 

1.96 

0.41 

2 

3 

2.94 

0.57 

2.94 

0.59 

2.94 

0.60 

2.94 

0.61 

3 

4 

3.93 

0.76 

3.92 

0.78 

3.92 

0.80 

3.92 

0.82 

4 

5 

4.91 

0.95 

4.90 

0.98 

4.90 

.00 

4.90 

1.02 

5 

6 

5.89 

1.14 

5.88 

1.17 

5.88 

.20 

5.87 

1.22 

6 

7 

6.87 

1.34 

6.87 

1.37 

6.86 

.40 

6.85 

1.43 

7 

8 

7.85 

1.53 

7.85 

1.56 

7.84 

.59 

7.83 

1.63 

8 

9 

8.83 

1.72 

8.83 

1.76 

8.82 

.79 

8.81 

1.83 

9 

10 

9.82 

1.91 

9.81 

1.95 

9.80 

.99 

9.79 

2.04 

10 

11 

10.80 

2.10 

10.79 

2.15 

10.78 

2.19 

10.77 

2.24 

11 

12 

11.78 

2.29 

11.77 

2.34 

11.76 

2.39 

11.75 

2.44 

12 

13 

12.  f6 

2.48 

12.75 

2.54 

12.74 

2.59 

12.73 

2.65 

13 

14 

13.74 

2.67 

13.73 

2.73 

13.72 

2.79 

13.71 

2.85 

14 

15 

14.72 

2.86 

14.71 

2.93 

14.70 

2.99 

14.69 

3.06 

15 

16 

15.71 

3.05 

15.69 

3.12 

15.68 

3.19 

15.66 

3.26 

16 

17 

16.69 

3.24 

16.67 

3.32 

16.66 

3.39 

16.64 

3.46 

17 

18 

17.67 

3.43 

17.65 

3.51 

17.64 

3.59 

17.62 

3.66 

18 

19 

18.65 

3.63 

18.63 

3.71 

18.62 

3.79 

18.60 

3.87 

19 

20 

19.63 

3.82 

19.62 

3.90 

19.60 

3.99 

19.58 

4.07 

20 

21 

20.61 

4.01 

20.60 

4.10 

20.58 

4.19 

20.56 

4.28 

21 

22 

21.60 

4.20 

21.58 

4.29 

21.56 

4.39 

21.54 

4.48 

22 

23 

22.58 

4.39 

22.56 

4.49 

22.54 

4.59 

22.52 

4.68 

23 

24 

23.56 

4.58 

23.54 

4.68 

23.52 

4.78 

23.50 

4.89 

24 

25 

24.54 

4.77 

24.52 

4.88 

24.50 

4.98 

24.48 

5.09 

25 

26 

25.52 

4.96 

25.50 

5.07 

25.48 

5.18 

25.46 

5.30 

26 

27 

26.50 

5.15 

26.48 

5.27 

26.46 

5.38 

26.43 

5.50 

27 

28 

27.49 

5.34 

27.46 

5.46 

27.44 

5.58 

27.41 

5.70 

28 

29 

28.47 

5.53 

28.44 

5.66 

28.42 

5.78 

28.39 

5.91 

29 

30 

29.45 

5.72 

29.42 

5.85 

29.40 

5.98 

29.37 

6.11 

30 

31 

30.43 

5.92 

30.40 

6.05 

30.38 

6.18 

30.35 

6.31 

31 

32 

31.41 

6.11 

31.39 

6.24 

31.36 

6.38 

31.33 

6.52 

32 

33 

32.39 

6  30 

32.37 

6.44 

32.34 

6.58 

32.31 

6.72 

33 

34 

33.38 

6.49 

33.35 

6.63 

33.32 

6.78 

33.29 

6.92 

34 

35 

34.36 

6.68 

34.33 

6.83 

34.30 

6.98 

34.27 

7.13 

35 

36 

35.34 

6.87 

35.31 

7.02 

35.28 

7.18 

35.25 

7.33 

36 

37 

36.32 

7.06 

36.29 

7.22 

36.26 

7.38 

36.22 

7.53 

37 

38 

37.30 

7.25 

37.27 

7.41 

37.24 

7.58 

37.20 

7.74 

38 

39 

38.28 

7.44 

38.25 

7.61 

38.22 

7.78 

38.18 

7.94 

39 

40 

39.27 

7.63 

39.23 

7.80 

39.20 

7.97 

39.16 

8.15 

40 

41 

40.25 

7.82 

40.21 

8.00 

40.18 

8.17 

40.14 

8.35 

41 

42 

41.23 

8.01 

41.19 

8.19 

41.16 

8.37 

41.12 

8.55 

42 

43 

42.21 

8.20 

42.17 

8.39 

42.14 

8.57 

42.10 

8.76 

43 

44 

43.19 

8.40 

43.15 

8.58 

43.12 

8.77 

43.08 

8.96 

44 

45 

44.17 

8.59 

44.14 

8.78 

44.10 

8.97 

44.06 

9.16 

45 

46 

45.15 

8.78 

45.12 

8.97 

45.08 

9.17 

45.04 

9.37 

46 

47 

46.14 

8.97 

46.10 

9.17 

46.06 

9.37 

46.02 

9.57 

47 

48 

47.12 

9.16 

47.08 

9.36 

47.04 

9.57 

46.99 

9.78 

48 

49 

48.10 

9.35 

48.06 

9.56 

48.02 

9.77 

47.97 

9.98 

49 

50 

49.08 

9.54 

49.04 

9.75 

49.00 

9.97 

48.95 

10.18 

50 

49 

o 
a 

Dep. 

Lai, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

o 

a 

ei 

.a 

Q 

79  Deg. 

78|  Deg. 

78£  Deg. 

78i  Deg, 

cA 

s 

TRAVERSE    TABLE. 


25 


3 

11  Deg. 

Hi  Deg, 

11£  Deg. 

11|  Deg, 

s 

So" 

£ 

3 

n 

0 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

§ 

~sT 

50.06 

9.73 

50.02 

9.95 

49.98 

10.17 

49.93 

10.39 

51 

52 

51.04 

9.92 

51.00 

10.14 

50.96 

10.37 

50.91 

10.59 

52 

53 

52.03 

10.11 

51.98 

10.34 

51.94 

10.57 

51.89 

10.79 

53 

54 

53.01 

10.30 

52.96 

10.53 

52.92 

10.77 

52.87 

11.00 

54 

55 

53.99 

10.49 

53.94 

10.73 

53.90 

10.97 

53.85 

11.20 

55 

56 

54.97 

10.69 

54.92 

10.93 

54.88 

11.16 

54.83 

11.40 

56 

57 

55.95 

10.88 

55.90 

11.12 

55.86 

11.36 

55.81 

11.61 

57 

53 

56.93 

11.07 

56.89 

11.32 

56.84 

11.56 

56.78 

11.81 

58 

59 

57.92 

11.26 

57.87 

11.51 

57.82 

11.76 

57.76 

12.01 

59 

60 

58.90 

11.45 

58.85 

11.71 

58.80 

11.96 

58.74 

12.22 

60 

61 

59.88 

11.64 

59.83 

11.90 

59.78 

12.16 

59.72 

12.42 

61 

62 

60,86 

11.83 

60.81 

12.10 

60.76 

12.36 

60.70 

12.63 

62 

63 

61,84    12.02 

61.79 

12.29 

61.74 

'12.56 

61.68 

12.83 

63 

64 

62.82 

12.21 

62.77 

12.49 

62.72 

12.76 

62.66 

13.03 

64 

65 

63.81 

12.40 

63.75 

12.68 

63.70 

12.96 

63.64 

13.24 

65 

66 

64,79 

12.59 

64.73 

12.88 

64.68 

13.16 

64.62 

13.44 

66 

67 

65,77 

12.78 

65.71 

13.07 

65.66 

13.36 

65.60 

13.64 

67 

63 

66.75 

12.98 

66.69 

13.27 

66.63 

13.56 

66.58 

13.85 

68 

69 

67.73 

13.17 

67.67 

13.46 

67.61 

13.76 

67.55 

14.05 

69 

70 

68.71 

13.36 

68.66 

13.66 

68.59 

13.96 

68.53 

14.25 

70 

71 

69.70 

13.55 

69.64 

13.85 

69.57 

14.16 

69.51 

14.46 

71 

72 

70.68 

13.74 

70.62 

14.05 

70.55 

14.35 

70.49 

14.66 

72 

73 

71.66 

13.93 

71.60 

14.24 

71.53 

,'4.55 

71.47 

14.87 

73 

74 

72.64 

14.12 

72.58 

14.44 

72.51 

14,75 

72.45 

15.07 

74 

75 

73.62 

14.31 

73.56 

14.63 

73.49 

14.95 

73.43 

15.27 

75 

76 

74.60 

14.50 

74.54 

14.83 

74.47 

15.15 

74.41 

15.48 

76 

77 

75.59 

14.69 

75.52 

15.02 

75.45 

15..  35 

75.39 

15.68 

77 

78 

76.57 

14.83 

76.50 

15.22 

76.43 

15".  55 

76.37 

15.88 

78 

79 

77.55 

15.07 

77.48 

15.41 

77.41 

15.75 

77.34 

16.09 

79 

_80 

78  .  53 

15.26 

78.46 

15.61 

78.39 

15.95 

78.32 

16.29 

80 

81 

79.51 

15.46 

79.44 

15.80 

79.37 

16.15 

79.30 

16.49 

81 

82 

80.49 

15.65 

80,42 

16.00 

80.35 

16.35 

80.28 

16.70 

82 

83 

81.48 

15.84 

81.41 

16.191 

81.33 

16.55 

81.26 

16.90. 

83 

84 

82.46 

16.03 

82.39 

16.39 

82.31 

16.75 

82.24 

17.11 

84 

85 

83.44 

16.22 

83.37 

16.53 

83.29 

16.95 

83.22 

17.31 

85 

86 

84.42 

16.41 

84.35 

16.78 

84.27 

17..  15 

84.20 

17.51 

86 

87 

85.40 

16.60 

85.33 

16.97 

85.25 

17.35 

85.18 

17.72 

87 

88 

86.38 

16.79 

86.31 

17.17 

86.23 

17.54 

86.16 

17.92 

88 

89 

87.36 

16,98 

87.29 

17.36 

87.21 

17.74 

87.14 

18.12 

89 

90 

88,35 

17.17 

88.27 

17.56 

88.19 

17.94 

88.11 

18.33 

90 

91 

89.33 

17.36 

89.25 

17.75 

89.17 

18.14 

89.09 

18.53 

91 

92 

90,31 

17.55 

90.23 

17.95 

90.15 

18.34 

90.07 

18.74 

92 

93 

91.29 

17.75 

91.21 

18.14 

91.13 

18.54 

91.05 

18.94 

93 

94 

92,27 

17.94 

92.19 

18.34 

92.11 

18.74 

92.03 

19-.  14 

94 

95 

93.25 

18.13 

93.17 

18.53 

93.09 

18.94 

93.01 

19.35 

95 

96    94.24 

18.32 

94.16 

18.73 

94.07 

19.14 

93.99 

19.55 

96 

97 

95.22 

18.51 

95.14 

18.92 

95.05 

19.34 

94.97 

19.75 

97 

98 

96.20 

18.70 

96.12 

19.12 

96.03 

19.54 

95.95 

19.96 

98 

99 

97.18 

18.89 

97.10 

19.31 

97.01 

19.74 

96.93 

20.1-6 

99 

100. 

93.16 

19.03 

98.08 

19.51 

97.99 

19.94 

97.90 

20.36 

jOO 

I 

Dep. 

Lat. 

Dep. 

Lat.  I 

Dep. 

Lat. 

Dep. 

Lat. 

1 

5 

1 

Q 

79  Deg. 

78|  Deg.     l      78±Deg. 

II 

78*  Deg. 

Q 

TRAVERSE   TABLE. 


s 

12  Deg. 

12k  Deg. 

m  Deg. 

12|  Deg. 

O 

00 

1 

I* 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

r 

0.98 

0.21 

0.98 

0.21 

0.98 

0.22 

0.98 

0.22 

1 

2 

1.96 

0.42 

1.95 

0.42 

1.95 

0.43 

1.95 

0.44 

2 

3 

2,93 

0.62 

2.93 

0.64 

2.93 

0.65 

2.93 

0.66 

3 

4 

3.91 

0.83 

3.91 

0.85 

3.91 

0.87 

3.90 

0.88 

4 

5 

4.89 

1.04 

4.89 

1.06 

4.88 

1.08 

4.88 

1.10 

5 

6 

5.87 

1.25 

5.86 

1.27 

5.86 

1.30 

5.85 

1.32 

6 

7 

6.85 

1.46 

•6.84 

1.49 

6.83 

1.52 

6.83 

1.54 

7 

8 

7.83 

1.66 

7.82 

1.70 

7.81 

1.73 

7.80 

1.77 

8 

9 

8.80 

1.87 

8.80 

1.91 

8.79 

1.95 

8.78 

1.99 

9 

10 

9.78 

2.08 

9.77 

2.12 

9.76 

2.16 

9.75 

2.21 

10 

11 

10.76 

2.29 

10.75 

2.33 

10.74 

2.38 

10.73 

2.43 

11 

12 

11.74 

2.49 

11.73 

2.55 

11.72 

2.60 

11.70 

2.65 

12 

13 

12.72 

2.70 

12.70 

2.76 

12.69 

2.81 

12.68 

2.87 

13 

14 

13.69 

2.91 

13.68 

2.97 

13.67 

3.03 

13.65 

3.09 

14 

15 

14.67 

3.12 

14.66 

3.18 

14.64 

3.25 

14.63 

3.31 

15 

16 

15.65 

3.33 

15.64 

3.39 

15.62 

3.46 

15.61 

3.53 

16 

17 

16.63 

3.53 

16.61 

3.61 

16.60 

3.68 

16.58 

3.75 

17 

18 

17.61 

3.74 

17.59 

3.82 

17.57 

3.90 

17.  C6 

3.97 

18 

19 

18.58 

3.95 

18.57 

4.03 

18.55 

4.11 

18.53 

4.19 

19 

20 

19.56 

4.16 

19.54 

4.24 

19.53 

4.33 

19.51 

4.41 

20 

21 

20.54 

4.37 

20.52 

4.46 

20.50 

4.55 

20.48 

4.63 

21 

22 

21.52 

4.57 

21.50 

4.67 

21.48 

4.76 

21.46 

4.86 

22 

23 

22.50 

4.78 

22.48 

4.88 

22.45 

4.98 

22.43 

5.08 

23 

24 

23.48 

4.99 

23.45 

5.09 

23.43 

5.19 

23.41 

5.30 

24 

25 

24.45 

5.20 

24.43 

5.30 

24.41 

5.41 

24.38 

5.52 

25 

26 

25.43 

5.41 

25.41 

5.52 

25.38 

5.63 

25.36 

5.74 

26 

27 

26.41 

5.61 

26.39 

5.73 

26.36 

5.84 

26.33 

5.96 

27 

28 

27.39 

5.82 

27.36 

5.94 

27.34 

6.06 

27.31 

6.18 

28 

29 

28.37 

6.03 

28.34 

6.15 

28.31 

6.28 

28.28 

6.40 

29 

30 

29.34 

6.24 

29.32 

6.37 

29.29 

6.49 

29.26 

6.62 

30 

31 

30  .  32 

6.45 

30.29 

6.58 

30.27 

6.71 

30.24 

6.84 

31 

32 

31.30 

6.65 

31.27 

6.79 

31.24 

6.93 

31.21 

7.06 

32 

33 

32.28 

6.86 

32.25 

7.00 

32.22 

7.14 

32.19 

7.28 

33 

34 

33.26 

7.07 

33.23 

7.21 

33.19 

7.36 

33.16 

7.50 

34 

35 

34.24 

7.28 

34.20 

7.43 

34.17 

7.58 

34.14 

7.72 

35 

36 

35.21 

7.48 

35.18 

7.64 

35.15 

7.79 

35.11 

7.95 

36 

37 

36.19 

7.69 

36.16 

7.85 

36.12 

8.01 

36.09 

8.17 

37 

38 

37.17 

7.90 

37.13 

8.06 

37.10 

8.22 

37.06 

8.39 

38 

39 

38.15 

8.11 

38.11 

8.27 

38.08 

8.44 

38.04 

8.61 

39 

40 

39.13 

8.32 

39.09 

8.49 

39.05 

8.66 

39.01 

8.83 

40 

41 

40.10 

8.52 

40.07 

8.70 

40.03 

8.87 

39.99 

9.05 

41 

42 

41.08 

8.73 

41.04 

8.91 

41.00 

9.09 

40.96 

9.27 

42 

43 

42.06 

8.94 

4?.  02 

9.12 

41.98 

9.31 

41.94 

9.49 

43 

44 

43.04 

9.15 

43.00 

9.34 

42.96 

9.52 

42.92 

9.71 

44 

45 

44.02 

9.36 

43.98 

9.55 

43.93 

9.74 

43.89 

9.93 

45 

46 

44.99 

9.56 

44.95 

9.76 

44.91 

9.96 

44.87 

10.15 

46 

47 

45.97 

9.77 

45.93 

9.97 

45.89 

10.17 

45.84 

10.37 

47 

48 

46.95 

9.98 

46.91 

10.18 

46.86 

10.39 

46.82 

10.59 

48 

49 

47.93 

10.19 

47.88 

10.40 

47.84 

10.61 

47.79 

10.81 

49 

50 

48.91 

10.40 

48.86 

10.61 

48.81 

10.82 

48  .  77 

11.03 

50 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0 

§ 

Q 

78  Deg. 

77|  Deg. 

771  Deg. 

774  Deg. 

rf 

s 

TRAVERSE    TABLE. 


27 


e 

12  Deg. 

12i  Deg. 

12£  Deg. 

12J  Deg. 

S 

£ 

1 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5i 

49.89 

10.00 

49.84 

10.82 

49.79 

11.04 

49.74 

11.26 

51 

52 

50.86 

10.81 

50.82 

11.03 

50.77 

11.25 

50.72 

11.48 

52 

53 

51.84 

11.02 

51.79 

11.25 

51.74 

11.47 

51.69 

11.70 

53 

54 

52.82 

11.23 

52.77 

11.46 

J52.72 

11.69 

52.67 

11.02 

54 

55 

53.80 

11.44 

53.75 

11.67 

153.70 

11.90 

53.64 

12.14 

55 

56 

54.78 

11.64 

54.72 

11.88 

J54.67 

12.12 

54.62 

12.36 

56 

57 

55.75 

11.85 

55.70 

12.09 

55.65 

12.34 

55.59 

12.58 

57 

58 

56.73 

12.06 

56.68 

12.31 

56.63 

12.55 

56.57 

12.80 

58 

59 

57.71 

12.27 

57.66 

12.52 

57.60 

12.77 

57.55 

13.02 

59 

60 

58.69 

12.47 

58.63 

12.73 

58.58 

12.99 

58.52 

13.24 

60 

61 

59.67 

12.68 

59.61 

12.94 

159.55 

13.20 

59.50 

13.46 

61 

62 

60.65 

12.89 

60.59 

13.16 

60.53 

13.42 

60.47 

13.68 

62 

63 

61.62 

13.10 

61.57 

13.37 

61   51 

13.64 

61.45 

13.90 

63 

64 

62.60 

13.31 

62.54 

13.58 

62  .'48 

13.85 

62.42 

14.12 

64 

65 

63.58 

13.51 

63.52 

13.79 

i63.46 

14.07 

63.40 

14.35 

65 

66 

64.56 

13.72 

64.50 

14.00 

J64.44 

14.29 

64.37 

14.57 

06 

67 

65.54 

13.93 

65.47 

14.22 

165.41 

14.50 

65.35 

14.79 

67 

68 

66.51 

14.14 

66.45 

14.43 

66.39 

14.72 

66.32 

15.01 

68, 

69 

67.49 

14.35 

67.43 

14.64 

67.36 

14.93 

67.30 

15.23 

69 

70 

68.47 

14.55 

68.41 

14.85 

68.34 

15.15 

68.27 

15.45 

70 

71 

69.45 

14.76 

69.38 

15.06 

69.32 

15.37 

69.25 

15.67 

71 

72 

70.43 

14.97; 

70.36 

15.28 

70.29 

15.58 

70.22 

15.89 

72 

73 

71.40 

15.18! 

71.34 

15.49 

71.27 

15.80 

71.20 

16.11 

73 

74 

72.38 

15.39 

72.32 

15.70 

72.25 

16.02 

72.18 

16.33 

74 

75 

73.36 

15.59 

73.29 

15.91 

73.22 

16.23 

73.15 

16.55 

75 

76 

74.34 

15.80 

74.27 

16.13 

74.20 

16.45 

74.13 

16.77 

76 

77 

75.32 

16.01 

75.25 

16.34 

75.17 

16.67 

75.10 

16.99 

77 

78 

76.30 

16.22 

76.22 

16.55 

76.15 

16.88 

76.08 

17.21 

78 

79 

77.27 

16.43 

77.20 

16.76 

77.13 

17.10 

77.05 

17.44 

79 

80 

78.25 

16.63 

78.18 

16.97 

78.10 

17.32 

78.03 

17.66 

80 

81; 

79.23 

16.84 

79.16 

17.19 

79.08 

17.53 

79.00 

17.88 

81 

82 

80.21 

17.05 

80.13 

17.40 

80.06 

17.75 

79.98 

18.10 

82 

83 

81.19 

17.26 

81.11 

17.61 

81.03 

17.96 

80.95 

18.32 

83 

84 

82.16 

17.46 

82.09 

17.82 

82.01 

18.18 

81.93 

18.54 

84 

85 

83.14 

17.67 

83.06 

18.04 

82.99 

18.40 

82.90 

18.76 

85 

86 

84.12 

17.  8S 

84.04 

18.25 

83.96 

18.61 

83.88 

18.98 

86 

87 

85.10 

18.09 

85.02- 

18.46 

84.94 

18.83 

84.85 

19.20 

87 

88 

86.08 

18.30 

86.00 

18.67 

85.91 

19.05 

85.83 

19.42 

88 

89 

87.06 

18.50 

86.97 

18.88 

86.89 

19.26 

86.81 

19.64 

89 

90 

88.03 

18..71 

87.95 

19.10 

87.87 

19.48 

87.78 

19.86 

90 

91 

89.01 

18.92 

88.93 

19.31 

88.84 

19.70 

88.76 

20.08 

PI 

92 

89.99 

19.13 

89.91 

19.52 

89.82 

19.91 

89.73 

20.30 

92 

93    90.97 

19.34 

90.88 

19.73 

90.80 

20.13 

90.71 

20.52 

9? 

94 

91.95 

19.54 

91.86 

19.94 

91.77 

20.35 

91.68 

20.75 

94 

95 

92.92 

19.75 

92.84 

20.16 

92.75 

20.56 

92.66 

20.97 

95 

96    93.90 

19.96 

93.81 

20.37 

93  .  72 

20.78 

93.63 

21.19 

96 

97    94.88 

20.17 

94.79 

20.08 

94.70 

20.99! 

94.61 

21.41 

97 

98  195.  86 

20.38 

95.77 

20.79 

95.68 

21.21 

95.58 

21.63 

98 

99 

96.  >4 

20.58 

96.75 

21.01 

96.65 

21.43 

96.56 

21.85 

99 

100 

97.81 

20.79 

97.72 

21.22 

97.63 

21.64 

97.53 

22.07 

100 

o 
o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

c 

rt 

1 

5 

78  Deg. 

77J  Deg 

771  Deg. 

77*  Deg. 

3 

M 


26 


TRAVERSE    TABLE. 


G 

13  Deg. 

13*  Deg. 

131  Deg. 

13J  Deg. 

C 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

0.97 

0.23 

0.97 

0.23 

0.97 

0.23 

0.97 

0.24 

1 

2 

1.95 

0.45 

1.95 

0.46 

1.95 

0.47 

1.94 

0.48 

2 

3 

2.92 

0.67 

2.92 

0.69 

2.92 

0.70 

2.91 

0.71 

3 

4 

3.90 

0.90 

3.89 

0.92 

3.89 

0.93 

3.89 

0.95 

4 

6 

4.87 

1.12 

4.87 

1.15 

4.86 

1.'17 

4.86 

1.19 

5 

6 

5.85 

1.35 

5.84 

1.38 

5.83 

1.40 

5.83 

1.43 

6 

7 

6.82 

1.57 

6.81 

1.60 

6.81 

1.63 

6.80 

1.66 

7 

8 

7.80 

1.80 

7.79 

1.83 

7.78 

1.87 

7.77 

1.90 

8 

9 

8.77 

2.02 

8.76 

2.06 

8.75 

2.10 

8.74 

2.14 

9 

10 

9.74 

2.25 

9.73 

2.29 

9.72 

2.33 

9.71 

2.38 

10 

11 

10.72 

2.47 

10.71 

2.52 

10.70 

2.57 

10.68 

2.61 

11 

12 

11.69 

2.70 

11.68 

2.75 

11.67 

2.80 

11.66 

2.85 

12 

13 

12.67 

2.92 

12.65 

2.98 

12.64 

3.03 

12.63 

3.09 

13 

14 

13.64 

3.io 

13.63 

3.21 

13.61 

3.27 

13.60 

3.33 

14 

15 

14.62 

3.37 

14.60 

3.44 

14.59 

3.50 

14.57 

3.57 

15 

16 

15.59 

3.60 

15.57 

3.67 

15.56 

3.74 

15.54 

3.80 

16 

17 

16.57 

3.82 

16.55 

8.90 

16.53 

3.97 

16.51 

4.04 

17 

18 

17.54 

4.05 

17.52 

4.13 

17.50 

4.20 

17.48 

4.28 

18 

19 

18.51 

4.27 

18.49 

4.35 

18.48 

4.44 

18.46 

4.52 

19 

20 

19.43 

4.50 

19.47 

4.58 

19.45 

4.67 

19.43 

4.75 

20 

21 

20.46 

4.72 

20.44 

4.81 

20.42 

4.90 

20.40 

4.99 

21 

22 

21.44 

4.95 

21.41 

5.04 

21.39 

5.14 

21.37 

5.23 

22 

23 

22.41 

5.17 

22.39 

5.27 

22.36 

5.37 

22.34 

5.47 

23 

24 

23.38 

5.40 

23.36 

5.50 

23.34 

5.60 

23.31 

5.70 

24 

25 

24.36 

5.62 

24.33 

5.73 

24.31 

5.84 

24.28 

5.94 

25 

26 

25.33 

5.85 

25.31 

5.96 

25.28 

6.07 

25.25 

6.18 

26 

27 

26.31 

6.07 

26.28 

6.19 

26.25 

6.3-0 

26.23 

6.42 

27 

28 

27.28 

6.30 

27.25 

6.42 

27.23 

6.54 

27.20 

6.66 

28 

29 

28.26 

6.52 

28.23 

6.65 

28.20 

6.77 

28.17 

6.89 

29 

30 

29.23 

6.75 

29.20 

6.88 

29.17 

7.00 

29.14 

7.13 

30 

31 

30.21 

6.97 

30.17 

7.11 

30.14 

7.24 

30.11 

7.37 

31 

32 

31.18 

7.20 

31.15 

7.33 

31.12 

7.47 

31.08 

7.61 

32 

33 

32.15 

7.42 

32.12 

7.56 

32.09 

7.70 

32.05 

7.84 

33 

34 

33.13 

7.65 

33.09 

7.79 

33.06 

7.94 

33.03 

8.08 

34 

35 

34.10 

7.87 

34.07 

8.02 

34.03 

8.17 

34  00 

8.32 

35 

36 

35.08 

8.10 

35.04 

8.25 

35.01 

8.40 

34.97 

8.56 

36 

37 

36.05 

8.32 

36.02 

8.48 

35.98 

•  8.64 

35.94 

8.79 

37 

38 

37.03 

8.55 

36.99 

8.71 

36.95 

8.87 

36.91 

9.03 

38 

39 

38.00 

8.77 

37.96 

8.94 

37.92 

9.10 

37.88 

9.27 

39 

40 

38.97 

9.00 

38.94 

9.17 

38.89 

9.34 

38.85 

9.51 

40 

41 

39.95 

9.22 

39.91 

9.40 

39.87 

9.57 

39.83 

9.75 

41 

42 

40.92 

9.45 

40.88 

9.63 

40.84 

9.80 

40.80 

9.98 

42 

43 

41.90 

9  67 

41.86 

9.86 

41.81 

10.04 

41.77 

10.22 

43 

44 

42.87 

9.90 

42.83 

19.08 

42.  73 

10.27 

42.74 

10.46 

44 

45 

43.85 

10.12 

43.80 

10.31 

43.76 

10.51 

43.71 

10.70 

45 

46 

44.82 

10.35 

44.78 

10.54 

44.73 

10.74 

44.68 

10.93 

46 

47 

45.80 

10.57 

45.75 

10.77 

45.70 

10.97 

45.65 

11.17 

47 

48 

46.77 

10.80 

46.72 

11.00 

46.67 

11.21 

46.62 

11.41 

48 

49 

47.74 

11.02 

47.70 

11.23 

47.65 

11.44 

47.60 

11.65 

49 

50 

48.72 

11.25 

48.67 

11.46 

48.62 

11.67 

48.57 

11.88 

50 

B 

o 

JS 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

ctf 
I 

5 

77  Deg. 

76|  Deg. 

76]  Deg. 

76*  Deg. 

ri 

.2 

b 

UNIVERSITY 

•  • 


TRAVERSE    TABLE. 


29 


c 

£' 

13  Deg. 

134  Deg. 

13*  Deg. 

13|  Deg. 

O 

Lance.l 

o 
? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

49.69 

11.47 

49.64 

11.69 

49.59 

11.91 

49.54 

12.12 

"51 

52 

50.67 

11.70 

50.62 

11.92 

50.56 

12.14 

50.51 

12.36 

52 

53 

51.64 

11.92 

51.59 

12.15 

51.54 

12.37 

51.48 

12.60 

53 

54 

52.62 

12.15 

52.56 

12.38 

52.51 

12.61 

52.45 

12.84 

54 

55 

53.59 

12.37 

53.54 

12.61 

53.48 

12.84 

53.42 

13.07 

55 

56 

54.56 

12.60 

54.51 

12.84 

54.45 

13.07 

54.40 

13.01 

56 

57 

55.54 

12.82 

55.48 

13.06 

55.43 

13.31 

55.37 

13.55 

57 

58 

56.51 

13.05 

56.46 

13.29 

56.40 

13.54 

56  34 

13.79 

58 

59 

57.49 

13.27 

57.43 

13.52 

57.37 

13.77 

57.31 

14.02 

59 

60 

58.46 

13.50 

58.40 

13.75 

58.34 

14.01 

58.28 

14.26 

60 

61 

59.44 

13.72 

59.38 

13.98 

59.31 

14.24 

59.25 

14.50 

61 

62 

60.41 

13.95 

60.35 

14.21 

60.29 

14.47 

60.22 

14.74 

62 

63 

61.39 

14.17 

61.32 

14.44 

61.26 

14.71 

61.19 

14.97 

63 

64 

62.36 

14.40 

62.30 

14.67 

62.23 

14.94 

62.17 

15.21 

64 

65 

63.33 

14.62 

63.27 

14.90 

63.20 

15.17 

63.14 

15.45 

65 

66 

64.31 

14.85 

64.24 

15.13 

64.18 

15.41 

64.11 

15.69 

66 

67 

65.28 

15.07 

65.22 

15.36 

65.15 

15.64 

65.08 

15.93 

67 

68 

66.26 

15.30 

66.19 

15.59 

66.12 

15.87 

66.05 

16.16 

68 

69 

67.23 

15.52 

67.16 

15.81 

67.09 

16.11 

67.02 

16.40 

69 

70 

68.21 

15.75 

68.14 

16.04 

68.07 

16.34 

67.99 

16.64 

70 

71' 

69.18 

15.97! 

69.11 

16.27 

69.04 

16.57 

68.97 

16.88 

71 

72 

70.15 

16.20 

70.08 

16.50 

70.01 

16.81 

69.94 

17.11 

72 

73 

71.13 

16.42 

71.06 

16.73 

70.98 

17.04 

70.91 

17.35 

73 

74 

72.10 

16.65 

72.03 

16.96 

71.96 

17.28 

71.88 

17.59 

74 

75 

73.08 

16.87 

73.00 

17.19 

72.93 

17.50 

72.85 

17.83 

75 

76 

74.05 

17.10 

73.98 

17.42 

73.90 

17.74 

73.82 

18.06 

76 

77 

75.03 

17.32 

74.95 

17.65 

74.87 

17.98 

74.79 

18.30 

77 

78 

76.00 

17.55 

75.92 

17.88 

75.84 

18.21 

75.76 

18.54 

78 

79 

76.98 

17.77 

76.90 

18.11 

76.82 

18.44 

76.74 

18.78 

79 

80 

77.95 

18.00 

77.87 

18.34 

77.79 

18.68 

77.71 

19.01 

80 

81 

78.92 

18.22 

78.84 

18.57 

78.76 

18.91- 

78.68 

19.25 

81 

82 

79.90 

18.45 

79.82 

18.79 

79.73 

19.14 

79.65 

19.49 

82 

83 

80.87 

18.67 

80.79 

19.02 

80.71 

19.38 

80.62 

19.73 

83 

84 

81.85 

18.90 

81.76 

19.25 

81.68 

19.61 

81.59 

19.97 

& 

85 

82.82 

19.12 

82.74 

19.48 

82.65 

19.84 

82.56 

20.20 

85 

86 

83.80 

19.35 

83.71 

19.71 

83.62 

20/08 

83.54 

20.44 

86 

87 

84.77 

19.57 

84.68 

19.94 

84.60 

20.31 

84.51 

20.68 

87 

88 

85.74 

19.80 

85.66 

20.17 

85.57 

20.54 

85.48 

20.92 

88 

89 

86.72 

20.02 

86.63 

20.40 

86.54 

20.78 

86.45 

21.15 

89 

90 

87.69 

20  .  25 

87.60 

20.63  187.51 

21.01 

87.42 

21.39 

90 

91 

88.67" 

20.47 

88.58 

20.86     88.49 

21.24 

88.39 

21.63 

91 

92 

89.64 

20.70 

89.55 

21.09     89.46 

21.48 

89.36 

21.87 

92 

93 

90.62 

20.92 

90.52 

21.32  190.43 

21.71 

90.33 

22.10 

93 

94 

91.59 

21.15 

91.50 

21.54    91.40 

21.94 

91.31 

22.34 

94 

95 

92.57 

21.37 

92.47 

21.77    92.38 

22.18 

92.28 

22.58 

95 

96 

93.54 

21.60 

93.44 

22.00     93.35 

22.41 

93.25 

22.82 

96 

97 

94.51 

21.82 

94.42 

22.23     94.32 

22.64 

94.22 

23.06 

97 

98 

95.49 

22.05 

95.39 

22.46     95.29 

22-88 

95.19 

23.29 

98 

99 

96.46 

22.27 

96.36 

22.69     96.26 

23.11 

96.16 

23.53 

99 

100 

97.44 

22.50 

97.34 

22.92     97.24 

23.34 

97.13 

23.77 

100 

• 
o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

rt 

5 

77  Deg. 

76J  Deg. 

76^  Deg. 

76*  Deg. 

cd 

51 

30 


TRAVERSE    TABLE. 


O 
sr 

— 

14  Deg. 

14i  Deg. 

14£  Deg. 

14|  Deg. 

O 
P 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

i 

0.97 

0.24 

0.97 

0.25 

0.97 

0.25 

0.97 

0.25 

1 

2 

1.94 

0.48 

1.94 

0.49 

1.94 

0.50 

1.93 

6.51 

2 

3 

2.91 

0.73 

2.91 

0.74 

2.  -90 

0.75 

2.90 

0.76 

3 

4 

3.88 

0.97 

3.88 

0.98 

3.87 

1.00 

3.87 

1.02 

4 

5 

4.S5 

1.21 

4.85 

1.23 

4.84 

1.25 

4.84 

1.27 

5 

6 

5.82 

1.45 

5.82 

1.48 

5.81 

1.50 

5.80 

1.53 

6 

7 

6.79 

1.69 

6.78 

1.72 

6.78 

1.75 

6.77 

1.78 

7 

8 

7.76 

1.94 

7.75 

1.97 

7.75 

2.00 

7.74 

2.04 

8 

9 

8.73 

2.18 

8.72 

2.22 

8.71 

2.25 

8.70 

2.29 

9 

10 

9.70 

2.42 

9.69 

2.46 

9.63 

2.50 

9.67 

2.55 

10 

11 

10.67 

2.66 

10.66 

2.71 

10.65 

2.75 

10.64 

2.80 

11 

12 

11.64 

2.90 

11.63 

2.95 

11.62 

3.00 

11.60 

3.06 

12 

13 

12.61 

3.15 

12.60 

3.20 

12.59 

3.25 

12.57 

3.31 

13 

14 

13.58 

3.39 

13.57 

3.45 

13.55 

3.51 

13.54 

3.56 

14 

15 

14.55 

3.63 

14.54 

3.69  1 

14.52 

3.76 

14.51 

3.82 

15 

16 

15.52 

3.87, 

15.51 

3.94 

15.49 

4.01 

15.47 

4.07 

16 

17 

16.50 

4.11 

16.43 

4.18! 

16.46 

4.26 

16.44 

4.33 

17 

18 

17.47 

4.35 

17.45 

4.43. 

17.43 

4.51 

17.41 

4.58 

18 

19 

18.44 

4.60 

18.42 

4.68  ' 

18.39 

4.76 

18.37 

4.84 

19 

20 

19.41 

4.84 

19.38 

4.92  1 

19.36 

5.01 

19.34 

5.09 

20 

21 

20.38 

5.08 

20.35 

5.17 

20.33 

5.26 

20.31 

5.35 

21 

22 

21.35 

5.32 

21.32 

5.42 

21.30 

5.51 

21.28 

5.00 

22 

23 

22.32 

5.56 

22.29 

5.66 

22.27 

5.76 

22.24 

5.86 

23 

24 

23.99 

5.81 

23.26 

5.91 

23.24 

6.01 

23.21 

6.11 

24 

25 

24.26 

6.05 

24.23 

6.15 

24.20 

6.26 

24.18 

6.37 

25 

26 

25.23 

6.29 

25.20 

6.40 

25.17 

6.51 

25.14 

6.62 

26 

27 

26.20 

6.53 

26.17 

6.65 

26.14 

6.76 

26.11 

6.87 

27 

28 

27.17 

6.77 

27.14 

6.89 

27.11 

7.01 

27.08 

7.13 

28 

29 

23.14 

7.02 

28.11 

7.14 

28.08 

7.26 

28.04 

7.38 

29 

30 

29.11 

7.26 

29.08 

7.38 

29.04 

7.51 

29.01 

7.64 

30 

31 

30.08 

7.50 

30.05 

7.63 

30.01 

7.76 

29.93 

7.89 

31 

32 

31.05 

7.74 

31.02 

7.88 

30.98 

8.01 

30.95 

8.15 

32 

33 

32.02 

7.98 

31.98 

8.12 

31.95 

8.26 

31.91 

8.40 

33 

34 

32.99 

8.23 

32,95 

8.37 

32.92 

8.51 

32.88 

8.66 

34 

35 

33.96 

8.47 

33.92 

8.62 

33.89 

8.76 

33.85 

8.91 

35 

36 

34.93 

8.71 

34.89 

8.86 

34.85 

9.01 

34.81 

9.17 

36 

37 

35.90 

8.95 

35.86 

9.11 

35.82 

9.26 

35.78 

9.42 

37 

38 

36.87 

9.19 

36.83 

9.35 

36.79 

9.51 

36.75 

9.67 

38 

39 

37  .  84 

9.44 

37.80 

9.60 

37.76 

9.76 

37.71 

9.93 

39 

40 

38.81 

9.68 

38.77 

9.85 

38.73 

10.02 

38.68 

10.18 

40 

41 

39.78 

9.92 

39.74 

10.09 

39.69 

10.27 

39.65 

10.44 

41 

42 

40.75 

10.16 

40.71 

10.34 

40.66 

10.52 

40.62 

10.69 

42 

43 

41.72 

10.40 

41.68 

10.58 

41.63 

10.77 

41.58 

10.95 

43 

44 

42.69 

10.64 

42.65 

10.83 

42.60 

11.02 

42  .  55 

11.20 

44 

4n 

43.66 

10.89 

43.62 

11.08 

43.57 

11.27 

43.52 

11.46 

45 

46 

44.63 

11.13 

44.58 

11.32 

44.53 

11.52 

44.48 

11.71 

46 

47 

45.60  11.37 

45.55 

11.57 

45.50 

11.77 

45.45 

11.97 

47 

48 

46.57 

11.61 

46.52 

11.82 

46.47 

12.02 

46.42 

12.22 

48 

49 

47.54 

11.85 

47.49 

12.06 

47.44 

12.27 

47.39 

12.48 

49 

50 

48.51 

12.10 

48.46 

12.31 

48.41 

12.52 

48.35 

12.73 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

s 

1 

1 

1C 

CO 

|5 

76  Deg. 

75|  Deg. 

7fii  Deg. 

75i  Deg. 

3 

TRAVERSE    TABLE. 


31 


b 

E 

14  Deg. 

144  Deg. 

14A  Deg. 

14|  Deg. 

? 

1 
9 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

51 

49.49 

12.34 

49.43 

12.55 

J49.  38 

12.77 

49.32 

12.98 

51 

52 

50.46 

12.58 

50.40 

12.80 

50.34 

13.02 

50.29 

13.24 

52 

53 

51.43 

12.82 

51.37 

13.05 

51.31 

13.27 

51.25 

13.49 

53 

54 

52.40 

13.06 

52.34 

13.29 

52.28 

13.52 

52.22 

13.75 

54 

55 

53.37 

13.31 

53.31 

13.54 

53.25 

13.77 

53.19 

14.00 

55 

56    54.34 

13.55 

54.28 

13.78 

54.22 

14.02 

54.15 

14.26 

56 

57    55.31 

13.79 

55.25 

14.03 

65.18 

14.27 

55.12 

14.51 

57 

58 

56.28 

14.03 

56.22 

14.28 

56.15 

14.52 

56.09 

14.77 

58 

59 

57.25 

14.27 

57.18 

14.52 

57.12 

14.77 

57.06 

15.02 

59 

60 

58.22 

14.52 

58.15 

14.77 

58.09 

15.02 

58  .  02 

15.28 

60 

61 

59.19 

14.76 

59.12 

15.02 

59.06 

15.27 

58.99 

15.53 

61 

62 

60.16 

15.00 

60.09 

15.26 

60.03 

15.52 

59.96 

15.79 

62 

63 

61.13 

15.24 

61.06 

15.51 

60.99 

15.77 

60.92 

16.04 

63 

64 

62.10 

15.48 

62.03 

15.75 

61.96 

16.02 

61.89 

16.29 

64 

65 

63.07 

15.72 

63.00 

16.00 

62.93 

16.27 

62.86 

16.55 

65 

66 

64.04 

15.97 

63.97 

16.25 

63.90 

16.53 

63.83 

16.80 

66 

67 

65.01 

16.21 

'64.94 

16.49 

64.87 

16.78 

64.79 

17.06 

67 

68 

65.98 

16.45 

65.91 

16.74 

65.83 

17.03 

65.76 

17.31 

68 

69 

66.95 

16.69 

66.88 

16.98 

66.80 

17.28 

66.73 

17.57 

69 

70 

67.92 

16.93 

67.85 

17.23 

67.77 

17.53 

67.69 

17.82 

70 

71 

68.89 

17.18 

68  .  82 

17.48 

68.74 

17.78 

68.66 

18.08 

71 

72 

69.86 

17.42 

69.78 

17.72 

69.71 

18.03 

69.63 

18.33 

72 

73 

70.83 

17.66 

70.75 

17.97 

70.67 

18.28 

70.59 

18.59 

73 

74 

71.80 

17.90 

71.72 

18.22 

71.64 

18.53 

71.56 

18.84 

74 

75 

72.77 

18.14 

72.69 

18.46 

72.61 

18.78 

72.53 

19.10 

75 

76 

73.74 

18.39 

73.66 

18.71 

73.58 

19.03 

73.50 

19.35 

76 

77 

74.71 

18.63 

74.63 

18.95 

74.55 

19.28 

74.46 

19.60 

77 

78 

75.68 

18.87 

75.60 

19.20 

75.52 

19.53 

75.43 

19.86 

78 

79 

76.65 

19.11 

76.57 

19.45 

76.48 

19.78 

76.40 

20.11 

79 

80 

77.62 

19.35 

77.54 

19.69 

77.45 

20.03 

77.36 

20.37 

80 

81 

78.59" 

19.60 

78.51 

19.94 

78.42 

20.28 

78.33 

20.62 

81 

82 

79.56 

19.84 

79.48 

20.18 

79.39 

20.53 

79.30 

20.88 

82 

83 

80.53 

20.08 

80.45 

20.43 

80.36 

20.78 

80.26 

21.13 

83 

84 

81.50 

20.32 

81.42 

20.68 

81.32 

21.03 

81.23 

21.39 

84 

85 

82.48 

20.56 

82.38 

20.92 

82.29 

21.28 

82.20 

21.64 

85 

86 

83.45 

20.81 

83.35 

21.17 

83.26 

21.53 

83,17 

21.90 

86 

87 

84.42 

21.05 

84.32 

21.42 

84.23 

21.78 

84.13 

22.15 

87 

88 

85.39 

21.29 

85.5:9 

21.66 

85.20 

22.03 

85.10 

22.41 

88 

89 

86.36 

21.53 

86.26 

21.91 

86.17 

22.28 

86.07 

22.66 

89 

90 

87.33 

21.77 

87.23 

22.15 

87.13 

22.53 

87.03 

22.91 

90 

91 

88.30 

22.01 

88  .20 

22.40 

88.10 

22.78 

88.00 

23.17 

91 

92 

89.27 

22.26 

89.17 

22.65 

89.07 

23.04 

88.97 

23.42 

92 

93 

90.24 

22.50 

90.14 

22.89 

90.04 

23.29 

89.94 

23.68 

93 

94  I  91.  21 

22.74 

91.11 

23.14 

91.01 

23.54 

90.90 

23.93 

94 

95192.18 

22.98 

92.08 

23.38 

91.97 

23.79 

91.87 

24.19 

95 

96    93.15 

23.22 

93.05 

23.63 

92.94 

24.04 

92.84 

24.44 

96 

97  '  94.12 

23.47 

94.02 

23.88;  93.91 

24.29 

93.80 

24.70 

97 

98    95.09 

23.71 

94.98 

24.12! 

94.88 

24.54 

94.77 

24.95 

98 

99    96.06 

23  9i, 

95.95 

24.37! 

95.85 

24.79 

95.74 

25.21 

99 

100    97.03    24.19 

96.92 

24.62  ,  96.81 

25.04 

96.70 

25.46 

100 

I 

Dep. 

Lat. 

Dcp. 

Lat.   |   Dep. 

Lat. 

Dep. 

Lat. 

1 

• 
P 

76  Deg. 

II 
75jDeg.     I       75^  Deg. 

S 
75*  Deg.      5- 

TRAVERSE   TABLE. 


2 

15  Deg. 

15i  Deg. 

15£  Deg. 

15|  Deg. 

O 
£" 
£T 

i 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

i 

0.97 

0.26 

0.96 

0.26 

0.96 

0.27 

0.96 

0.27 

1 

2 

1.93 

0.52 

1.93 

0.53 

1.93 

0.53 

1.92 

0.54 

2 

3 

2.90 

0.78 

2.89 

0.79 

2.89 

0.80 

2.89 

0.81 

3 

4 

3.86 

1.04 

3.86 

1.05 

3.85 

1.07 

3.85 

1.09 

4 

5 

4.83 

1.29 

4.82 

1.32 

4.82 

1.34 

4.81 

1.36 

5 

6 

5.80 

1.55 

5.79 

1.58 

5.78 

1.60 

5.77 

1.63 

6 

7 

6.76 

1.81 

6.75 

1.84 

6.75 

1.87 

6.74 

1.90 

7 

8 

7.73 

2.07 

7.72 

2.10 

7.71 

2.14 

7:70 

2.17 

8 

9 

8.69 

2.33 

8.68 

2.37 

8.67 

2.41 

8.66 

2.44 

9 

10 

9.66 

2.59 

9.65 

2.63 

9.64 

2.67 

9.62 

2.71 

10 

11 

10.63 

2.85 

10.61 

2.89 

10.60 

2.94 

10.59 

2.99 

11 

12 

11.59 

3.11 

11.58 

3.16 

11.56 

3.21 

11.55 

3.26 

12 

13 

12.56 

3.36 

12.54 

3.42 

12.53 

3.47 

12.51 

3.53 

13 

14 

13.52 

3.62 

13.51 

3.68 

13.49 

3.74 

13.47 

3.80 

14 

15 

14.49 

3.88 

14.47 

3.95 

14.45 

4.01 

14.44 

4.07 

15 

16 

15.45 

4.14 

15.44 

4.21 

15.42 

4.28 

15.40 

4.34 

16 

17 

16.42 

4.40 

16.40 

4.47 

16.38 

4.54 

16.36 

4.61 

17 

18 

17.39 

4.66 

17.37 

4.73 

17.35 

4.81 

17.32 

4.89 

18 

19 

18.35 

4.92 

18.33 

5.00 

18.31 

5.08 

18.29 

5.16 

19 

20 

19.32 

5.18 

19.30 

5.26 

19.27 

5.34 

19.25 

5.43 

20 

21 

20.28 

5.44 

20.26 

5.52 

20.24 

5.61 

20.21 

5.70 

21 

22 

21.25 

5.69 

21.23 

5.79 

21.20 

5.88 

21.17 

5.97 

22 

23 

22.22 

5.95 

22.19 

6.05 

22.16 

6.15 

22.14 

6.24 

23 

24 

23.18 

6.21 

23.15 

6.31 

23.13 

6.41 

23.10 

6.51 

24 

25 

24.15 

6.47 

24.12 

6.58 

24.09 

6.68 

24.06 

6.79 

25 

26 

25.11 

6.73 

25.08 

6.84 

25.05 

6.95 

25.02 

7.06 

26 

27 

26.08 

6.99 

26.05 

7.10 

26.02 

7.22 

25.99 

7.33 

27 

28 

27.05 

7.25 

27.01 

7.36 

26.98 

7.48 

26.95 

7.60 

28 

29 

28.01 

7.51 

27.98 

7.63 

27.95 

7.75 

27.91 

7.87 

29 

30 

28.98 

7.76 

28.94 

7.89 

28.91 

8.02 

28.87 

8.14 

30 

31  '29.94 

8.02 

29.91 

8.15 

29.87 

8.28 

29.84 

8.41 

31 

32  30.91 

8.28 

30.87 

8.42 

30.84 

8.55 

30.80 

8.69 

32 

33 

31.88 

8.54 

31.84 

8.68 

31.80 

8.82 

31.76 

8.96 

33 

34 

32.84 

8.80 

32  ,  80 

8.94 

32.76 

9.09 

32.72 

9.23 

34 

35 

33.81 

9.06 

33.77 

9.21 

33.73 

9.35 

33.69 

9.50 

35 

36 

34.77 

9.32 

34.73 

9.47 

34.69 

9.62 

34.65 

9.77 

36 

37 

35.74 

9.58 

35.70 

9.73 

35.65 

9.89 

35.61 

10.04 

37 

38 

36.71 

9.84 

36.66 

10.00 

36  .  62 

10.16 

36.57 

10.31 

38 

39 

37.67 

10.09 

37.63 

10.26 

37.58 

10.42 

37.54 

10.59 

39 

40 

38.64 

10.35 

38.59 

10.52 

38.55 

10.69 

38.50 

10.86 

40 

41 

39.60 

10.61 

39.56 

10.78 

39.51 

10.96 

39.46 

11.13 

41 

42 

40.57 

10.87 

40.52 

11.05 

.40.47 

11.22 

40.42 

11.40 

42 

43 

41.53 

11.13 

41.49 

11.31 

41.44 

11.49 

41.39 

11.67 

43 

44 

42.50 

11.39 

42.45 

11.57 

42.40 

11.76 

42.35 

11.94 

44 

45 

43.47 

11.65 

43.42 

11.84 

43.36 

12.03 

43.31 

12.21 

45 

46 

44.43 

11.91 

44.38 

12.10 

44.33 

12.29 

44.27 

12.49 

46 

47 

45.40 

12.16 

45.35 

12.36 

45.29 

12.56 

45.24 

12.76 

47 

48 

46.36 

12.42 

46.31 

12.63 

46.25 

12.83 

46.20 

13.03 

48 

49 

47.33 

12.68 

47.27 

12.89 

47.22 

13.09 

47.16 

13.30 

49 

50 

48.30 

12.94 

48.24 

13.15 

48.18 

13.36 

48.12 

13.57 

50 

8 

C 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

9 

O 

c 

1 

rt 
w 

i 

75  Deg. 

74}  I>eg. 

74>-  Deg. 

74i  Deg. 

s 

Tfi AVERSE    TABLE. 


c 

1 
P 

15  Deg. 

15*  Deg. 

« 

Deg. 

15|  Deg. 

5 
P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 
52 
53 
54 

ii 

57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

49.26 
50.23 
51.19 
52.16 
53.13 
54.09 
55.06 
56.02 
56.99 
57.96 

I3.20l 
13.46 
13.72 
13.98 
14.24 
14.49 
14.75i 
15.01 
15.27 
15.53 

49.20 
50.17 
51.13 
52.10 
53.06 
54.03 
54.99 
55.96 
56.92 
57.89 

13.41 
13.68 
13.94 
14.20 
14.47 
14.73 
14.99 
15.26 
15.52 
15.78 

49,15 
50.11 
51.07 
52.04 
53.00 
53.96 
54.93 
55.89 
56.85 
57.82 
58.78 
59.75 
60.71 
61.67 
62.64 
63.60 
64.56 
65.53 
66.49 
67.45 

13.63 
13.90 
14.16 
14.43 
14.70 
14.97 
15.23 
15.50 
15.77 
16.03 

49.09 
50.05 
51.01 
51.97 
52.94 
53.90 
54.86 
55.82 
56.78 
57.75 

13.84 
14.11 
14.39 
14.66 
14.93 
15.20 
15.47 
15.74 
16.01 
16.29 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

58.92 
59.89 
60.85 
61.82 
62.79 
63.76 
64.72 
65.68 
66.65 
67.61 

15.79 
16.05 
16.31 
16.56 
16.82 
17.08 
17.34 
17.60 
17.86 
18.12 

58.85 
59.82 
60.78 
61.75 
62.71 
63.68 
64.64 
65.61 
66.57 
67.54 

16.04 
16.31 
16.57 
16.83 
17.10 
17.36 
17.62 
17.89 
18.15 
18.41 

16.30 
16.57 
16.84 
17.10 
17.37 
17.64 
17.90 
18.17 
18.44 
18.71 

58.71 
59.67 
60.63 
61.60 
62.56 
63.52 
64.48 
65.45 
66.41 
67.37 

16.56 
16.83 
17.10 
17.37 
17.64 
17.92 
18.19 
18.46 
18.73 
19.00 

61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

71 
72 
73 
74 
75 
76_ 

68.58 
69.55 
70.51 
71   48 

18.38 
18.63 
18.89 
19.15 
19.41 
19.67 

68.50 
69.46 
70.43 
71.39 
72.36 
73.JJ2- 
74.29 
75.25 
76.22 
77.18 

18.68 
18.94 
19.20 
19.46 
19.73 
19.99 
20.25 
20.52 
20.78 
21.04 

68.42 
69.38 
70.35 
71.31 
72.27 
73.24 
74.20 
75.16 
76.13 
77.09 

18.97 
19.24 
19.51 
19.78 
20.04 
20.31 
20.58 
20.84 
21.11 
21.38 

68.33 
69.30 
70.26 
71.22 
72.18 
73.15 
74.11 
75.07 
76.03 
77.00 

19.27 
19.54 
19.82 
20.09 
20.36 
20.63 
20.90 
21.17 
21.44 
21.72 

71 
72 
73 

74 
75 
76 
77 
78 
79 
80 

Jf 

78 
79 

80 

74.38 
75.34' 
76.31 
77.27 

19.93 
20.19 
20.45 
20.71 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

78.24 
79.21 
80.17 
81.14 
82'.  10 
83.07 
84.04 
85.00 
85-.  97 
86.93 

20.96 
21.22 
21.48 
21.74 
22.00 
22.26 
22.52 
22.78 
23.03 
23.29 

78.15 
79.11 
80.08 
81.04 
82.01 
82.97 
83.94 
84.90 
85.87 
86.83 

21.31 
21.57 
21.83 
22.09 
22.36 
22.62 
22.88 
23.15 
23.41 
23.67 

78.05 
79.02 
79.98 
80.94 
81.91 
82.87 
83.84 
84.80 
85.76 
86.73 

21.65 
21.91 
22.18 
22.45 
22.72 
22?98 
23.25 
23.52 
23.78 
24.05 

77.96 
78.92 
79.88 
80.85 
81.81 
82.77 
83.73 
84.70 
85.66 
86.62 

21.99 
22.26 
22.53 
22.80 
23.07 
23.34 
23.62 
23.89 
24.16* 
24.43 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

91 
92 
93 
94 
95 
96 
97 
98 
99 
100 

G 

87.90 
88.87 
89.83 
90.80 
91.76 
92.73 
93.69 
94.66 
95.63 
96.59 

23.55 
23.81 
24.07 
24.33 
24.59 
24.85 
25.11 
25.36 
25.62 
25.88 

87.80 
88.76 
89.73 
90.69 
91.65 
92.62 
93.58 
94.55 
95.51 
96.48 

23.94 
24.20 
24.46 
24.72 
24.99- 
25.25 
25.51 
25.78 
26.04 
26.30 

87.69 
88.65 
89.62 
90.58 
91.54 
92.51 
93.47 
94.44 
95.40 
96.36 

24.32 
24.59 
24.85 
25.12 
25.39 
25.65 
25.92 
26.19 
26.46 
26.72 

87.58 
88.55 
89.51 
90.47 
91.43 
92.40 
93.36 
94.32 
95.28 
96.25 

24.70 
24.97 
25.24 
25.52 
25.79 
26.06 
26.33 
26.60 
26.87 
27.14 

91 
92 
93 
94 
95 
96 
97 
98 
99 
100 
to 

w 
To 

3 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

75  Deg. 

74*  Deg. 

741  Deg. 

744  Deg. 

34 


TRAVERSE   TABLE. 


•g 

16  Deg. 

16i  Deg, 

161  Deg. 

16|  Deg. 

C 

CO 

! 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

ft 

CD 

'i 

1)796" 

0.28 

0.96 

0.28 

0.96 

0.28 

0.96 

0.29 

~T 

2 

1  92 

0,55 

1.92 

0.56 

1.92 

0.57 

1.92 

0.58 

2 

3 

2.88 

0.83 

2.88 

0.84 

2.88 

0.85. 

2.87 

0.86 

3 

4 

3.85 

1.10 

3.84 

1.12 

3.84 

1.14 

3:83 

1  .  15 

4 

5 

4.81 

1.38 

4.80 

1.40 

4.79 

1.42 

4.79 

1.44 

5 

6 

5.77 

1.65 

5.76 

1.68 

5.75" 

1.70 

5.75 

1.73 

6 

7 

6.73 

1.93 

6.72 

1.96 

6.71 

1.99 

6.70 

2.02 

7 

8 

7.69 

2.21 

7.68 

2.24 

7.67 

2.27 

7.66 

2.31 

8 

9 

8.65 

2.48 

8.64 

2.52 

8.63 

2.56 

8.62 

2.59 

9 

10 

9.61 

2.76 

9.60 

2.80 

9.59 

2.84 

9.58 

2.88 

10 

11 

10.57 

3.031 

10.56 

3.08 

10.55 

3.12 

10.53 

3.17 

11 

12 

11.54 

3.31 

11.52 

3.36 

11.51 

3.41 

11.49 

3.46 

12 

13 

12.50 

3.58 

12.48 

3.64 

12.46 

3.69 

12.45 

3.75 

13 

14' 

13.46 

3.86 

13.44 

3.92 

13.42 

3.98 

13.41 

4.03 

14 

15 

14.42 

4.13 

14.40 

4.20 

14.38 

4.26 

14.36 

4.32 

15 

16 

15.38 

4.41 

15.36 

4.48 

15.34 

4.54 

15.32 

4.61 

16 

17 

16.34 

4.69 

16.32 

4.76 

16.30 

4.83 

16.28 

4.90 

17 

18 

17.30 

4.96 

17.28 

5.04 

17.26 

5.11 

17.24 

5.19 

18 

19 

18.26 

5.24 

18.24 

5.32 

18.22 

5.40 

18.19 

5.48 

19 

20 

19.23 

5.51 

19.20 

5.60 

19.18 

5.68 

19.15 

5.76 

20 

21 

20.19 

5.79! 

20.16 

5.88 

20.14 

5.96 

20.11 

6.05 

21 

22 

21.15 

6.06 

21.12 

6.16 

21.09 

6.25 

21.07 

6.34 

22 

23 

22.11 

6.34 

22.08 

6.44 

22.05 

6.53 

22.02 

6.63 

23 

24 

23.07 

6.62 

23.04 

6.72 

23.01 

6.82 

22.98 

6.92 

24 

25 

24.03 

6.89 

24.00 

7.00 

23.97 

7.10 

23.94 

7.20 

25 

26 

24.99 

7.17 

24.96 

7.28 

24.93 

.7.38 

24.90 

7.49 

26 

27 

25.95 

7.44 

25  .  92 

.7.56 

25.89 

r.tt 

25.85 

7.78 

'27 

28 

26.92 

7.72 

26.88 

7.84 

26.85 

7.95 

26.81 

8.07 

28 

29 

27.88 

7.99 

27.84 

8.11 

27.81 

8.24 

27.77 

8.36 

29 

30 

28.84 

8.27 

28.80 

8.39 

28.76 

8.52 

28.73 

8.65 

30 

31 

29.80 

8.54 

29  .  76 

8.67 

29  .  72 

8.80 

29.68 

8.93 

31 

32 

30.76 

8.82 

30.72 

8.95 

30.68 

9.09 

30.64 

9.22 

32 

33 

31.72 

9.10 

31.68 

9.23 

31.64 

9.37 

31.60 

9.51 

33 

34 

32.68 

9.37 

32.64 

9.51 

32.60 

9.66 

32.56 

9.80 

34 

35 

33.64 

9.65 

33.60 

9.79 

33.56 

9.94 

33.51 

10.09 

35 

36 

34.61 

9.92 

34.56 

10.07 

34.52 

10.22 

34.47 

10.38 

36 

37 

35.57 

10.20 

35.52 

10.35 

35.48 

10.51 

35.43 

10.66 

37 

38 

35.53 

10.47 

36.48 

10.63 

36.44 

10.79 

36.39 

10.95 

38 

39 

67.49 

10.75 

37.44 

10.91 

37.39 

11.08 

37.35 

11.  -24 

39 

40 

38.45 

11.03 

38.40 

11.19 

38.35 

11.36 

38.30 

11.53 

40 

41 

39.41 

11.30 

39.36 

11.47 

39.31 

11.64 

39.26 

11.82 

41 

42 

40.37 

11.58 

40.32 

11.75 

40.27 

11.93 

40.22 

12.10 

42 

43 

41.33 

11.85 

41.28 

12.03 

41.23 

12.21 

41.18 

12.39 

43 

44 

42.30 

12.13 

42.24 

12.31 

42.19 

12.50 

42.13 

12.68 

44 

45 

43.26 

12.40 

43.20 

12.59 

43.15 

12.78 

43.09 

12.97 

45 

46 

44.22 

12.68 

44.16 

12.87 

^4.11 

13.06 

44.05 

13.26 

46 

47 

45.18 

12.95 

45.12 

13.15 

45.08 

13.35 

45.01 

13.55 

47 

48 

46.14 

13.23 

46.08 

13.43 

46.02 

13.63 

45  .  96 

13.83 

48 

49 

47.10 

13.51 

47.04 

13.71 

46.98 

13.92 

46.92 

14.12 

49 

50 

43.06 

13.78 

48.00 

13.99 

47.94 

14.20 

47.88 

14.41 

50 

§ 

Dep. 

Lat. 

Dep. 

Lat 

Pep. 

Lat. 

Dep. 

Lat. 

V 

o 

c 

5 

74  Beg. 

73|  Deg. 

731  Deg. 

734  Deg. 

rf 

^ 

Q 

TRAVERSE    TABLE 


35 


d 

16  Deg. 

16}  Deg. 

16^  Deg. 

16|  Deg. 

- 

1 

? 

s 
a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

s 

n 

51 

49.02  \  14.06 

48.96 

14.27 

48.90 

14.48 

48.84 

14.70 

~5~I 

52 

49.99 

14.33 

49.92 

14.55 

49.86 

14.77 

49.79 

14.99 

52 

53 

50.95 

14.61 

50.88 

14.83 

50.82 

15.05 

50.75 

15.27 

53 

54 

51.91 

14.88 

51.84 

15.11 

51.78 

15.34 

51.71 

15.56 

54 

55 

52.87 

15.16 

52.80 

15.39 

52.74 

15.62 

52.67 

15.86 

55 

56 

53.83 

15.44 

53.76 

15.67 

53.69 

15.90 

53.62 

16.14 

56 

57 

54.79 

15.71 

54.72 

15.95 

54.65 

16.19 

54.58 

16.43 

57 

58 

55  .  75 

15.99! 

55.68 

16.23 

55.61 

16.47 

55.54 

16.72 

58 

59 

56.71 

16.J6     56.64 

16.51 

56.57 

16.76 

56.50 

17.00 

59 

60 

57.68 

16.54 

57.60 

16.79 

57.53 

17.04 

57.45 

17.29 

60 

61 

58.64 

16.81 

58.56 

17.07 

58.49 

17.32 

58.41 

17.58 

61 

62 

59.60 

17.09 

59.52 

17.35 

59.45 

17.61 

59.37 

17.87 

62 

63 

60.56 

17.37 

60.48 

17.63 

60.41 

17.89 

60.33 

18.16 

63 

64 

61.52 

17.64 

61.44 

17.91 

61.36 

18.18 

61.28 

18.44 

64 

65 

62.48 

17.92 

62.40 

18.19 

62.32 

18.46 

62.24 

18.73 

65 

66 

63.44 

18.19 

63.36 

18.47 

63.28 

18.74 

63.20 

19.02 

66 

67 

64.40 

18.47 

64.32 

18.75 

64.24 

19.03; 

64.16 

19.31 

67 

68 

65.37 

18.74 

65.28 

19.03 

65.20 

19.31 

65.11 

19.60 

68 

69    66.33 

19.02 

66.24 

19.31 

66.16 

19.60 

66.07 

19.89 

69 

70 

67.29 

19.29 

67.20 

19.59 

67.12 

19.88 

67.03 

20.17 

70 

71 

68.25 

19.57 

68.16 

19.87 

68.08 

20.17! 

67.99 

20.46 

71 

72 

69.21 

19.85 

69.12 

20.15 

69.03 

20.45 

68.95 

20.75 

72 

73 

70.17 

20.12 

70.08 

20.43 

69.99 

20.73 

69.90 

21.04 

73 

74 

71.13 

20.40 

71.04 

20.71 

70.95 

21.02 

70.86 

21.33 

74 

75 

72.09 

20.67 

72.00 

20.99 

71.91 

21.30 

71.82 

21.61 

75 

76 

73.06 

20.95 

72.96 

21.27 

72.87 

21.59 

72.78 

21.90 

76 

77 

74.02 

21.22 

73.92 

21.55 

73.83 

21.87 

73.73 

22.19 

77 

78 

74.98 

21.50 

74.88 

21.83 

74.79 

22.15 

74.69 

22.48 

78 

79 

75.94 

21.78 

75.84 

22.11 

75.75 

22.44 

7.5.65 

22.77 

79 

80 

76.90 

22.05 

76.80 

22.39 

76.71 

22.72 

76.61 

23.06 

80 

81 

77.86 

22.33 

77.76 

22.67 

77.66 

23.01 

77.56 

23.34 

81 

82 

78.82 

22.60 

78.72 

22.95 

78.62 

23.29 

78.52 

23.63 

82 

83 

79.78 

22.88 

79.68 

23.23 

79.58 

23.57 

79.48 

23.92 

83 

84 

80.75 

23.15 

80.64 

23.51 

80.54 

23.86 

80.44 

24.21 

84 

85 

81.71 

23.43 

81.60 

23.79 

81.50 

24.14 

81.39 

24.50 

85 

86 

82.67 

23.70 

82.56 

24.07 

82.46 

24.43 

82.35 

24.78 

86 

91 

83.63 

23.98 

83.52 

24.35 

83.42 

24,71 

83.31 

25.07 

87 

88 

84.59 

24.26 

84.48 

24.62 

84.38 

24.99 

184.27 

25.36 

88 

89 

85.55 

24.53 

85.44 

24.90 

85.33 

25.28 

85.22 

25.65 

89 

90 

8G.51 

24.81 

86.40 

25.18 

86.29 

25.56 

;86.18 

25.94 

90 

91 

87.47 

25.08 

87.36 

25.46 

87.25 

25.85 

|87.14 

26.23 

91 

92 

88.44 

25.36 

88.32 

25.74 

88.21 

26.13 

88.10 

26.51 

92 

93 

89.40 

25.63 

89.28 

26.02 

89.17 

26.41 

:89.05 

26.80 

93 

94 

90.36 

25.91 

90.24 

26.30 

90.13 

26.70 

90.01 

27.09 

94 

95 

91.32 

26.19 

91.20 

26.58 

91.09 

26.98 

90.97 

27.38 

95 

96 

92.28 

26.46 

92.16 

26.86 

92.05 

27.27 

91.93 

27.67 

96 

97 

93.24    26.74 

93.12    27.14 

93.01 

27.55 

92.88 

27.95 

97 

98 

94.20    27.01 

94.08 

27.42 

93.96 

27.83 

93.84 

28.24 

98 

99 

95.16    27.29 

95.04 

27.70 

94.92 

28.12 

94.80 

28.53 

99 

100 

96.13    27.56 

96.00    27.98 

95.88 

28.40 

95.76 

28.82 

100 

jj 
| 

Dep. 

Lat. 

Dep. 

Xat. 

Dep. 

Lat. 

Dep. 

Lat. 

g 

5 

1 

P 

74  Deg. 

73£  Deg. 

73iDeS. 

73i  Deg. 

Q 

N 


36 


TRAVERSE    TABLE. 


o 

t* 

17  Deg. 

174  Deg. 

17^  Deg. 

17|  Deg. 

o 

5- 

p» 

V 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D 
C5 
? 

1 

0.96 

0.29 

0.95 

0.30 

0.95 

0.30 

0.95 

0.30 

I 

2 

1.91 

0.58 

1.91 

0.59 

1.91 

0.60 

1.90 

0.61 

2 

3 

2.87 

0,88 

2.87 

0.89 

2.86 

0.90 

2.86 

0.91 

3 

4 

3.83 

1,17 

3.82 

1.19 

3.81 

1.20 

3.81 

1.22 

4 

5 

4.78 

1.46 

4.78 

1.48 

4.77 

1.50 

4.76 

1.52 

5 

6 

5.74 

1.75 

5.73 

1.78 

5.72 

1.80 

5.71 

1.83 

6 

7 

6.69 

2,05 

6.69 

2.08 

6.68 

2.10 

6.67 

2.13 

7 

8 

7.65 

2.34 

7.64 

2.37 

7.63 

2.41 

7.62 

2.44 

8 

9 

8.61 

2,63 

8.60 

2.67 

8.58 

2.71 

8.57 

2.74 

9 

10 

9.56 

2.92 

9.55 

2.97 

9.54 

3.01 

9.52 

3.05 

10 

11 

10.52 

3.22 

10.51 

3.26 

10.49 

3.31 

10.48 

3.35 

11 

12 

U.48 

3.51 

11.45 

3.56 

11.44 

3.61 

11.43 

3.66 

12 

13 

12.43 

3.80 

12.42 

3.85 

12.40 

3.91 

12.38 

3.96 

13 

14 

13.39 

4.09 

13.37 

4.15 

13.35 

4.21 

13.33 

4.27 

14 

15 

14.34 

4.39 

14,33 

4.45 

14.31 

4.51 

14.29 

4.57 

15 

16 

15.30 

4.68 

15.28 

4.74 

15.26 

4.81 

15.24 

4.88 

16 

17 

16.26 

4.97 

16.24 

5.04 

16.21 

5.11 

16.19 

5.18 

17 

18 

17.21 

5.26 

17.19 

5.34 

17.17 

5.41 

17.14 

5.49 

18 

19 

18.17 

5.56 

18.15 

5.63 

18.12 

5.71 

18.10 

5.79 

19 

20 

19.13 

5.85 

19.10 

5.93 

19.07 

6.01 

19.05 

6.10 

20 

21 

20.08 

6.14 

20.06 

6.23 

20.03 

6.31 

20.00 

6.40 

21 

22 

21.04 

6.43 

21.01 

6.52 

20.98 

6.62 

20.95 

6.71 

22 

23 

21.99 

6.72 

21.97 

6.82 

21.94 

6.92 

21.91 

7.01 

23 

24 

22.95 

7.021 

22.92 

7.12 

22.89 

7.22 

22.86 

7.32 

24 

25 

23.91 

7.31 

23.88 

7.41 

23.84 

7.52 

23.81 

7.62 

25 

26 

24.86 

7.60 

24.83 

7.71 

24.80 

7.82 

24.76 

7.93 

26 

27 

25.82 

7.89 

25.79 

8.01 

25.75 

8.12 

25.71 

8.23 

27 

28 

26.78 

8.19 

26.74 

8.30 

26.70 

8.42 

26.67 

8.54 

28 

29 

27.73 

8.48 

27.70 

8.60 

27.66 

8.72 

27.62 

8.84 

29 

30 

28.69 

8.77 

28.65 

8.90 

28.61 

9.02 

28.57 

9.15 

30 

31 

29.65 

9.06 

29.61 

9.19 

29.57 

9.32 

29.52 

9.45 

31 

32 

30.60 

9.36 

30.56 

9.49 

30.52 

9.62 

30.48 

9.76 

32 

33 

31.56 

9.65 

31.52 

9.79 

31.47 

9.92 

31.43 

10.06 

33 

34 

32.51 

9.94 

32.47 

10.08 

32.43 

10.22 

32.38 

10.37 

34 

35 

33.47 

10.23 

33.43 

10.38 

33.38 

10.52 

33.33 

10.67 

35 

36 

34.43 

10.53 

34.38 

10.68 

34.33 

10.83 

34.29 

tO.  98 

36 

37 

35.38    10.82 

35.34 

10.97 

35.29 

11.13 

35  24 

11.28 

37 

38 

36.34 

.11.11 

36  .  29 

11,27 

36.24 

11.43 

36.19 

11.58 

38 

39 

37.30 

11.40 

37.25 

11.57 

37.19 

11.73 

37.14 

11.89 

39 

40 

38.25*    11.69 

38.20 

11.86 

38.15 

12.03 

38.10 

12.19 

40 

41 

39.21 

11.99 

39.16 

12.16 

39.110 

12.33 

39.05 

12.50 

41 

42 

40.16 

12.28 

40  .  1  1 

12.45 

40.06 

12.63 

40.00 

12.80 

42 

43 

41.12 

12.57 

41.07 

12.75 

41.01 

12.93 

40.95 

13.11 

43 

44 

42.08 

12.86 

42.02 

13.05 

41.96 

13.23 

41.91 

13»41 

44 

45  |  43.03 

13.16 

42.98 

13.34 

42.92 

13.53 

42.86 

13.72 

45 

46 

43.99 

13.45 

43.93 

13.64 

43.87 

13.83 

43.81 

14,02 

46 

47 

44.95 

13.74 

44.89 

13.94 

44.82 

14.13 

44.76 

14.33 

47 

48 

45.90 

14.03 

45.84 

14.23 

45.78 

14.43 

45.71 

14.63 

48 

49 

46.86 

14.33 

46.80 

14.53 

46.73 

14.73 

46.67 

14.94 

49 

50 

47.82 

14.62 

47.75 

14.83 

47.69 

15.04 

47.62 

15.24 

50 

1 
3 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

V 

o 

c 

Q 

73  Deg. 

72|  Deg. 

721  Deg. 

72i  Deg. 

rf 

Q 

TRAVERSE    TABLE. 


37 


o 

17  Deg. 

17*  Deg. 

171  Deg. 

17|  De0r. 

O 

EC" 

p 

i 

1 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

~5*i 

48.77 

11.91 

48.71 

15.12 

48.64 

15.34 

48.57 

15.55 

TT 

62 

49.73 

15.20 

49.66 

15.42 

49.59 

15.64 

49.52 

15.85 

52 

53 

50.68 

15.50 

50.62 

15.72 

50.55 

15.94 

50.48 

16.16 

53 

54 

51.64 

15.79 

51.57 

16.01 

51.50 

16.24 

51.43 

16.46 

54 

55 

52.60 

16.08 

52.53 

16.31 

52.45 

16.54 

52.38 

16.77 

55 

56 

53.55 

16.37 

53.48 

16.61 

53.41 

16.84 

53.33 

17.07 

56 

57 

54.51 

16.67 

54.44 

16.90 

54.36 

17.14 

54.29 

17.38 

57 

58 

55.47 

16.96 

55.39 

17.20 

55.32 

17.44 

55.24 

17.68 

58 

59 

56.42 

17.25 

66.35 

17.50 

56.27 

17.74 

56.10 

17.99 

59 

60 

57.38 

17.54 

57.30 

17.79 

57.22 

18.04 

57.14 

18.29 

60 

61 

58.33 

17.83 

58.26 

18.09 

58.18 

18.34 

58.10 

18.60 

61 

62 

59.29 

18.13 

59.21 

18.39 

59.13 

18.64 

59.05 

18.90 

62 

63 

60.25 

18.42 

60.17 

18.68 

60.08 

18.94 

«0.00 

19.21 

63 

64 

61.20 

18.71 

61.12 

18.98 

61.04 

19.25 

60.95 

19.51 

64 

65 

62,16 

19.00 

62.08 

19.28 

61.99 

19.55 

61.91 

19.82 

65 

66 

63.12 

19.30 

63.03 

19.57 

62.95 

19.85 

62.86 

20.12 

66 

67 

64.07 

19.59 

63.99 

19.87 

63.90 

20.15 

63.81 

20.43 

67 

68 

65.03 

19.88 

64.94 

20.16 

64.85 

20.45 

64.76 

20.73 

68 

69 

65.99 

20.17;  65.90 

20.46 

65.81 

20.75 

65.72 

21.04 

69 

70 

6ft.  94 

20.47! 

66.85 

20.76 

66.76 

21.05 

66.67 

21.34 

70 

71 

67.90 

20.76 

67.81 

21.05 

67.71 

21.35 

67.62 

21.65 

71 

72 

68.85 

21.05 

68.76 

21.35 

68.67 

2L.66 

68.57 

21.95 

72 

73 

69.81 

21.34 

69.72 

21.65 

69.62 

21.95 

69.52 

22.26 

73 

74 

70.77 

21.64 

70.67 

21.94 

70.58 

22.25 

70.48 

22.56 

74 

75 

71.72 

21.93 

71.63 

22.24 

71.53 

22.55 

71.43 

22.86 

75 

76 

72.68 

22.22 

72.58 

22.54 

72.48 

22.85 

72.38 

23.17 

76 

77 

73.64 

22.51 

73.54 

22.83 

73.44 

23.15 

73.33 

23.47 

77 

78 

74.59 

22.80 

74.49 

23.13  1,74.39 

23.46 

74.29 

23.78 

78 

79 

75.55 

23.10 

75.45 

23.43  I  75.34 

23.76 

75.24 

24.08 

79 

80 

76.50 

23.39 

76.40 

23.72  '  76.30 

24.06 

76.19 

24.39 

80 

81 

77.46 

23.68 

77.36 

24.02     77.25 

24.36 

77.14 

24.69 

81 

82 

78.42 

23.97 

78.31 

24.32:178.20 

24.66 

78.10 

25.00 

82 

83 

79.37 

24.27 

79.27 

24.61  i 

79.16 

25.96 

79.05 

25.30 

83 

84 

80.33 

24.56 

80.22 

24.91  i 

80.11 

25.26 

80.00 

25.61 

84 

85 

81.29 

24.85 

81.18 

25.21  ' 

81.07 

25.56 

80.95 

25.91 

85 

86 

82.24 

25.14 

82.13 

25.50  : 

82.02 

25.86 

81.91 

26.22 

86 

87 

S3.  20 

25.44 

83.09 

25.80 

82.97 

26.16 

82.86 

26.52 

87 

88 

84.15 

25.73 

84.04 

26.10- 

83.93 

26.46 

83.81 

26.83 

88 

89 

85.11 

26.02 

85.00 

26.39  j 

84.88 

26.76 

84.76 

27.13 

89 

90 

86.07 

26.31 

85.95 

26.69 

85.83 

27.06 

85  .  7-2 

27.44 

90 

91 

87.02 

26.61 

86.  9i 

26.99 

86.79 

27.36 

86.67 

27.74 

91 

92 

87.98 

26.90 

87.86 

27.28 

87.  ^4 

27.66 

87.62 

28.05 

92 

93 

88.94 

27.19 

88.82 

27.58 

88.70 

27.97 

88.57 

28.35 

93 

94 

S9.89 

27.48 

89.77 

27.87  J89.65 

28.27 

89.53 

28.66 

94 

9o 

90.85 

27.78 

90.73 

28.17  190.60 

28.57 

90.48 

28.96 

95 

9!') 

91.81 

28.07 

91.68 

28.47  'J91.56 

28.87 

91.43 

29.27 

96 

97 

92.76 

28.36 

92.64 

28.76 

92.51 

29.17 

92.38 

29.57 

97 

98 

93.72 

28.65 

93.59 

29.06 

93.46 

29.47 

93.33 

29.88 

98 

99 

94.67 

28.94 

94.55 

29.36 

94.42 

29.77 

94.29 

30.18 

99 

100 

95.63 

29.24 

.95.50 

29.65 

95.37 

30.07 

95.24 

30.49 

100 

8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

rt 

Q 

73  Deg. 

72|  Deg. 

72j  Deg. 

72*  Deg. 

3 

38 


TRAVERSE    TABLE. 


»• 

18  Deg. 

184  Deg. 

18|  Deg. 

18|  Deg. 

O 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

B 
I 

1 

0.95 

0.31 

0.95 

0.31 

0.95 

0.32 

0.95 

0.32 

1 

2 

1.90 

0.62 

1.90 

0.63 

1.90 

0.63 

1.89 

0.64 

2 

3 

2.85 

0.93 

2.85 

0.94 

2.84 

0.95 

2.84 

0.96 

3 

4 

3.80 

1.24 

3.80 

1.25 

3.79 

1.27 

3.79 

1.29 

4 

5 

4.76 

1.55 

4.75 

1.57 

4.74 

1.59 

4.73 

1.61 

5 

6 

5.71 

1.85 

5.70 

1.88 

5.69 

1.90 

5.68 

1.93 

6 

7 

6.66 

2.16 

6.65 

2.19 

6.64 

2.22 

6.63 

2.25 

7 

8 

7.61 

2.47 

7.60 

2.51 

7.59 

2.54 

7.58 

2.57 

8 

•  9 

8.56 

2.78 

8.55 

2.82 

8.53 

2.86 

8.52 

2.89 

9 

10 

9.51 

3.09 

9.50 

3.13 

9.48 

3.17 

9.47 

3.21 

10 

11 

10.46 

3.40 

10.45 

3.44 

10.43 

3.49 

10.42 

3.54 

11 

12 

11.41 

3.71 

11.40 

3.76 

11.38 

3.81 

11.36 

3.86 

12 

13 

12.36 

4.02 

12.35 

4.07 

12.33 

4.12 

12.31 

4.18 

13 

14 

13.31 

4.33 

13.30 

4.38 

13.28 

4.44 

13.26 

4.50 

14 

15 

14.27 

4.64 

14.25 

4.70 

14.22 

4.76 

14.20 

4.82 

15 

16 

15.22 

4.94 

15.20 

5.01 

15.17 

5.08 

15.15 

5.14 

16 

17 

16.17 

5.25 

16.14 

5.32 

id.  12 

5.39 

16.10 

5.46 

17 

18 

17.12 

5.56 

17.09 

5.64 

17.07 

5.71 

17.04 

5.79 

18 

19 

18.07 

5.87 

18.04 

5.95 

18.02 

6.03 

17.99 

6.11 

19 

20 

19.02 

6.18 

18.99 

6.26 

18.97 

6.35 

18.94 

6.43 

20 

21 

19.97 

6.49 

19.94 

6.58 

19.91 

6.66 

19.89 

6.75 

21 

22 

20.92 

6.80 

20.89 

6.89 

20.86 

6.98 

20.83 

7.07 

23 

23 

21.87 

7.11 

21.84 

7.20 

21.81 

7.30 

21.78 

7.39 

23 

24 

22.83 

7.42 

22.79 

7.52 

22.76 

7.62 

22.73 

7.71 

24 

25 

23.78 

7.73 

23.74 

7.83 

23.71 

7.93 

23.67 

8.04 

25 

26 

24.73 

8.03 

24.69 

8.14 

24.66 

8.25 

24.62 

8.36 

26 

27 

25.68 

8.34 

25.64 

8.46 

25.60 

8.57 

25.57 

8.68 

27 

28 

26.63 

8.65 

26.59 

8.77 

26.55 

8.88 

26.51 

9.00 

28 

29 

27.58 

8.96 

27.54 

9.08 

27.50 

9.20 

27.46 

9.32 

29 

30 

28.53 

9.27 

28.49 

9.39 

28.45 

9.52 

28.41 

9.64 

30 

31 

29.48 

9.58 

29.44 

9.71 

29.40 

9.84 

29.35 

9.96 

31 

32 

30.43 

9.89 

30.39 

10.02 

30.35 

10.15 

30.30 

10.29  32 

33 

31.38 

10.20 

31.34 

10.33 

31.29 

10.47 

31.25 

10.61 

33 

34 

32.34 

JO.  51 

32.29 

10.65 

32.24 

10.79 

32.20 

10.93 

34 

35 

33.29 

10.82 

33.24 

10.96 

33.19 

11.11 

33.14 

11.25 

35 

36 

34.24 

11.12 

34.19 

11.27 

34.14 

11.42 

34.09 

11.57 

36 

37 

35.19 

11.43 

35.14 

11.59 

35.09 

11.74 

35.04 

11.89 

37 

38 

36.14 

11.74 

36.09 

11.90 

36,04 

12.00 

35.98 

12.21 

3S 

39 

37.09 

12.05 

37.04 

12.21 

36.98 

12.37 

36.93 

12.54 

39 

40 

38.04 

12.36 

37.99 

12.53 

37.93 

12.69 

37.88 

12.86 

40 

41 

38.99 

12.67 

38.94 

12.84 

38.88 

13.01 

38.82 

13.18 

41 

42 

39.94 

12.98 

39.89 

13.15 

39.83 

13.33 

39.77 

13.50 

42 

43 

40.90 

13.29 

40.84 

13.47 

40.78 

13.64 

40.72 

13.82 

43 

44 

41.85 

13.60 

41.79 

13.78 

41.73 

13.96 

41.66 

14.14 

44 

45 

42.80 

13.91 

42.74 

14.09 

42.67 

14.28 

42.61 

14.46 

45 

46 

43.75 

14.21 

43.69 

14.41 

43.62 

14.60 

43.56 

14.79 

46 

47 

44.70 

14.52 

44.64 

14.72 

44.57 

14.91 

44.51 

15.11 

47 

48 

45.65 

14.83 

45.59 

15.03 

45.52 

15.23 

45.45 

15.43 

48 

49 

46.60 

15.14 

46.54 

15.35 

46.47 

15.55 

46.40 

F5.75 

49 

50 

47.55 

15.4,6 

47.48 

15.66 

47.42 

15.87 

47.35 

16.07 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8) 

§ 

I 

72  Deg. 

71|  Deg. 

»HD* 

71  i  Deg. 

OB 

5 

TRAVERSE    TABLE. 


39 


1 

18  Deg. 

18i  Deg. 

18£  Deg. 

18|  Deg. 

O 

p 

Lat. 

Dep. 

Lat. 

•Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

51 

18.50 

15.76 

48.43 

15.97 

48.36 

16.18 

48.29 

16.39 

51 

52 

49.45 

16.07 

49.38 

16.28 

49.31 

16.50 

49.24 

16.71 

52 

53 

50.41 

16.38 

50.33 

16.60 

50.26 

16.82 

50.19 

17.04 

53 

54 

51.36 

16.69 

51.28 

16.91 

51.21 

17.13 

51.13 

17.36 

54 

55 

52.31 

17.00 

52.23 

17.22 

52.16 

17.45 

52.08 

17.68 

55 

56 

53.26 

17.30 

53.18 

17.54 

53.11 

17.77 

53.03 

18.00 

56 

57 

54.21 

17.61 

54.13 

17.85 

54.05 

18.09 

53.98 

18.32 

57 

58 

55.16 

17.92 

55.08 

18.16 

55.00 

18.40 

54.92 

18.64 

58 

59 

56.11 

18.23 

56.03 

18.48 

55.95 

18.72 

55.87 

18.96 

59 

60 

57.06 

18.54 

56.  9$ 

18.79 

56.90 

19.04 

56.82 

19.29 

60 

61 

58.01 

18  85 

57.93 

19.10 

57.85 

19.36 

57.76 

19.61 

61 

62 

58.97 

19.16 

58.88 

19.42 

58.80 

19.67 

58.71 

19.93 

62 

63 

59.92 

19.47 

59.83 

19.73 

59.74 

19.99 

59  .  66 

20.25 

63 

64 

60.87 

19.78 

60.78 

20.04 

60.69 

20.31 

60.60 

20.57 

64 

65 

61.82 

20.09 

61.73 

20.36 

61.64 

20.62 

61.55 

20.89 

65 

66 

62.77 

20.40 

62.68 

20.67 

62.59 

20.94 

62.50 

21.22 

66 

67 

63.72 

20.70 

63.63 

20.98 

63.54 

21.26 

63.44 

21.54 

67 

68 

64.67 

21.01 

64.58 

21.30 

64.49 

21.58 

64.39 

21.86 

68 

69 

65.62 

21.32 

65.53 

21.61 

65.43 

21.89 

65  .  34 

22.18 

69 

70 

66.57 

21.63 

66.48 

21.92 

66.38 

22.21 

66.29 

22.50 

70 

71 

67.53 

21.94 

67.43 

22.23 

67.33 

22  .  53 

67.23 

22.82 

71 

72 

68.48 

22.25 

68.38 

22.55 

68.28 

22.85 

68.18 

23.14 

72 

73 

69.43 

22.56 

69.33 

22.86 

69.23 

23.16 

69.13 

23.47 

73 

74 

70.38 

22.87 

70.28 

23.17 

70.18 

23.48 

70.07 

23.79 

74 

75 

71.33 

23.18 

71.23 

23.49 

71.12 

23.80 

71.02 

24.11 

75 

76 

72.28 

23.49 

72.18 

23.80 

72.07 

24.12 

71.97 

24.43 

76 

77 

73.23 

23.79 

73.13 

24.11 

73.02 

24.43 

72.91 

24.75 

77 

78 

74.18 

24.10 

74.08 

24.43 

73.97 

24.75 

73.86 

25.07 

78 

79 

75.13 

24.41 

75.03 

24.74 

74.92 

25.07 

74.81 

25.39 

79 

80 

76.08 

24.72 

75.98 

25.05 

75.87 

25.38 

75.75 

25.72 

80 

81 

77.04 

25.03 

76.93 

25.37 

76.81 

25.70 

76.70 

26.04 

81 

82 

77.99 

25.34 

77.88 

25.68 

77.76 

26.02 

77.65 

26.36 

82 

83 

78.94 

25.65 

78.83 

25.99 

78.71 

26.34 

78.60 

26.68 

83 

84 

79.89 

25.96 

79  .  77 

26.31 

79.66 

26.65 

79.54 

27.00 

84 

85 

80.84 

26.27 

80.72 

26.62 

80.61 

26.97 

80.49 

27.32 

85 

86 

81.79 

26.58 

81.67 

26.93 

81.56 

27.29 

81.44 

27.64 

86 

87 

82.74 

2&.S8 

82.62 

27.25 

82.50 

27.61 

82.38 

27.97 

87 

88 

83.69 

27.19 

83.57 

27.56 

83.45 

27.92 

83.33 

28.29 

88 

89 

84.64 

27.50 

84.52 

27.87 

84.40 

28.24 

84.28 

28.61 

89 

90 

85.60 

27.81 

85.47 

28.18 

85.35 

28.56 

85.22 

28.93 

90 

91 

§6.55 

28.12 

86.42 

28.50 

86.30 

28.37 

86.17 

29.25 

91 

92 

87.50 

28.43 

87.37 

28.81 

87.25 

29.19 

87.12 

29.57 

92 

93 

88.45 

28.74 

88.32 

29.12 

88.19 

29.51 

88.06 

29.89 

93 

94 

89.40 

29.05 

89.27 

29.44 

89.14 

29.83 

89.01 

30.22 

94 

95 

90.35 

29.36 

90.22 

29.75 

90.09 

30.14 

89.96 

30.54 

95 

96 

91.30 

29.67 

91.17 

30.06 

91.04 

30.46 

90.91 

30/86 

96 

97 

92.25 

29.97 

92.12 

30.38 

91.99 

30.78 

91.85 

31.18 

97 

98 

93.20 

30.28 

93.07!  30.69 

92.94 

31.10 

92.80 

31.50 

98 

99 

94.15 

30.59 

94.02 

31.00 

93.88 

31.41 

93.75 

31.82 

99 

100 

95.11 

30.90 

94.97 

31.32 

94.83 

31.73 

94.69 

32.14 

100 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

cd 

.2 

72  Deg. 

71J  Deg. 

71£  Deg. 

7H  Deg. 

" 

Q 

40 


TRAVERSE    TABLE. 


o 

5' 

19  Deg. 

19i  Deg. 

19£  Deg. 

19|  Deg. 

s 

•  § 
P 

ta 
P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.95 

0.33 

0.94 

0.33 

0.94 

0.33 

0.94 

0.34 

1 

2 

1.89 

0.65 

1.89 

0.66 

1.89 

0.67 

1.88 

0.68 

2 

3 

2.84 

0.98 

2.83 

0.99 

2.83 

1.00 

2.82 

1.01 

3 

4 

3.78 

1.30 

3.78 

1.32 

3.77 

1.34 

3.76 

1.35 

4 

5 

4.73 

1.63 

4.72 

1.65 

4.71 

1.67 

4.71 

1.69 

5 

6 

5.67 

1.95 

5.66 

1.98 

5.66 

2.00 

5.65 

2.03 

6 

7 

6.62 

2.28 

6.61 

2.31 

6.60 

2.34 

6.59 

2.37 

7 

8 

7.56 

2.60 

7.55 

2.64 

7.54 

2.67 

7.53 

2.70 

8 

9 

8.51 

2.93 

8.50 

2.97 

8.48 

3.00 

8.47 

3.04 

9 

10 

9.46 

3.26 

9.44 

3.30 

9.43 

•G.34 

,  9.41 

3.38 

10 

11 

10.40 

3.58 

10.38 

3.63 

10.37 

3.67 

10.35 

3.72 

11 

12 

11.35 

3.91 

11.33 

3.96 

11.31 

4.01 

11.29 

4.06 

12 

13 

12.29 

4.23 

12.27 

4.29 

12.25 

4.34 

12.24 

4.39 

•13 

14 

13.24 

4.56 

13.22 

4.62 

13.20 

4.67 

13.18 

4.73 

14 

15 

14.18 

4.88 

14.16 

4.95 

14.14 

5.01 

14.12 

5.07 

15 

16 

15.13 

5.21 

15.11 

5.28 

15.08 

5.34 

15.06 

5.41 

16 

17 

16.07 

5.53 

16.05 

5.60 

16.02 

5.67 

16.00 

5.74 

17 

18 

17.02 

5.86 

16.99 

5.93 

16.97 

6.01 

16.94 

6.08 

18 

19 

17.96 

6.19 

17.94 

6.26 

17.91 

6.34 

17.88 

6.42 

19 

20 

18.91 

6.51 

18.88 

6.59 

18.85 

6.68 

18.82 

6.76 

20 

21 

19.86 

6.84 

19.83 

6.92 

19.80 

7.01 

19.76 

7.10 

21 

22 

20.80 

7.16 

20.77 

7.25 

20.74 

7.34 

20.71 

7.43 

22 

23 

21.75 

7.49 

21.71 

7.58 

21.68 

7.68 

21.65 

7.77 

23 

24 

22.69 

7.81 

22.66 

7.91 

22.62 

8.01 

22.59 

8.11 

24 

25 

23.64 

8.14 

23.60 

8.24 

23.57 

8.35 

23.53 

8.45 

25 

26 

24.58 

8.46 

24.55 

8.57 

24.51 

8.68 

24.47 

8.79 

26 

27 

25.53 

8.79 

25.49 

8.90 

25.45 

9.01 

25.41 

9.12 

27 

28 

26.47 

9.12 

26.43 

9.23 

26.39 

9.35 

26.35 

9.46 

28 

29 

27.42 

9.44 

27.38 

9.56 

27.34 

9.68 

27.29 

9.80 

29 

30 

28.37 

9.77 

28.32 

9.89 

28.28 

10.01 

28.24 

10.14 

30 

31 

29.31 

10.09 

29.27 

10.22 

29.22 

10.35 

29.18 

10.48 

31 

32 

30.26 

10.42 

30.21 

10.55 

30.16 

10.68 

30.12 

10.81 

32 

33 

31.20 

10.74 

31.15 

10.88 

31.11 

11.02 

31.06 

11.15 

33 

34 

32.15 

11.07 

32.10 

11.21 

32.05 

11.35 

32.00 

11.49 

34 

35 

33.09 

11.39 

33.04 

11.54 

32.99 

11.68 

32.94 

11.83 

35 

36 

34.04 

11.72 

33.99 

11.87 

33.94 

12.02 

33.88 

12.17 

36 

37 

34.98 

12.05 

34.93 

12.20 

34.88 

12.35 

34.82 

12.50 

37 

38 

35.93 

12.37 

35.88 

12.53 

35.82 

12.68 

35.76 

12.84 

38 

39 

36.88 

12.70 

36.82 

12.86 

36.76 

13.02 

36.71 

13.18 

39 

40 

37.82 

13.02 

37.76 

13.19 

37.71 

13.35 

37.65 

13.52 

40 

41 

38.77 

13.35 

38.71 

13.52 

38.65 

13.69" 

38.59 

13.85 

41 

42 

39.71 

13.67 

39.65 

13.85. 

39.59 

14.02 

39.53 

14.19 

42 

43 

40.66 

14.00 

40.60 

14.18 

40.53 

14.35 

40.47 

14.53 

43 

44 

41.60 

14.32 

41.54 

14.51 

41.48 

14.69 

41.41 

14.87 

44 

45 

42.55 

14.65 

42.48 

14.84 

42,42 

15.02 

42.35 

15.21 

45 

46 

43.49 

14.98 

43.43 

15.17 

43.36 

15.36 

43.29 

15.54 

46 

47 

44.44 

15.30 

44.37 

15.50 

44.30 

15.69 

44.24 

15.88 

47 

48 

45.38 

15.63 

45.32 

15.83 

45.25 

16.02 

45.18 

16.22 

48 

49 

46.33 

15.95 

46.26 

16.15 

46.19 

16.36 

46.12 

16.56 

49 

50 

47.28 

16.28 

47.20 

16.48 

47.13 

16.69 

47.06 

16.90 

no 

8 

A 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 
o 

c 

d 

i 

5 

71  Deg. 

70}  Deg. 

70£  Deg. 

70}  Deg. 

s 

TRAVERSE    TABLE. 


41 


b 

19  Deg. 

19$  Deg. 

191  Dog. 

19|  Deg. 

o 

f 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

48.22 

16.60 

48.15 

16.81 

48.07 

17.02 

48.00 

17.23 

51 

52 

49.17 

16.93 

49.09 

17.14 

49.02 

17.36 

48.94 

17.57 

52 

53 

50.11 

17.26 

50.04 

17.47 

49.96 

.17.69 

49.88 

17.91 

53 

54 

51.06 

17.58 

50.98 

17.80 

50.90 

18.03 

50.82 

18.25 

54 

55 

52.00 

17.91 

51.92 

18.13 

51.85 

18.36 

51.76 

18.59 

55 

56 

52.95 

18.23 

52.87 

18.46 

52.79 

18.69 

52.71 

18.92 

56 

57 

53.89 

18.56 

53.81 

18.79 

53.73 

19.03 

53.65 

19.26 

57 

58 

54.84 

18.88 

54.76 

19.12 

54.67 

19.36 

54.59 

19.60 

58 

59 

55.79 

19.21 

55.70 

19.45 

55.62 

19.69 

55.53 

19.94 

59 

60 

56.73 

19.53 

56.65 

19.78 

56.56 

20.03 

56.47 

20.27 

60 

61 

57.68 

19.86 

57.59 

20.11 

57.50 

20.36 

57.41 

20.61 

61 

62 

58.62 

20.19 

58.53 

20.44 

58.44 

20.70 

58.35 

20,95 

62 

63 

59.57 

20.51 

59.48 

20.77 

59.39 

21.03 

59.29 

21.29 

63 

64 

60.51 

20.84 

60.42 

21.10 

60.33 

21.36 

60.24 

21.63 

64 

65 

61.46 

21.16 

61.37 

21.43 

61.27 

21.70 

61.18 

21.96 

65 

66 

62.40 

21.49 

62.31 

21.76 

62.21 

22.03 

62.12 

22.30 

66 

67 

63.35 

1-21.81 

63.25 

22.09 

63.16 

22.37 

63.06 

22.64 

67 

68 

64.  30  122.14 

64.20 

22.42 

64.10 

22.70 

64.00 

22.98 

68 

69 

65.24 

22.40 

65.14 

22.75 

65.04 

23.03 

64.94 

23.32 

69 

70 

66.19 

22.79 

66.09 

23.08 

65.98 

23.37 

65.88 

23.65 

70 

71 

67.13 

23.12 

67.03 

23.41 

66.93 

23.70 

66.82 

23.99 

71 

72 

68.03 

23.44 

67.97 

23.74 

67.87 

24.03 

67.76 

24.33 

72 

73 

69.02 

23.77 

68.92 

24.07 

68.81 

24.37 

68.71 

24.67 

73 

74 

69.97 

24.09 

69.86 

24.40 

69.76 

24.70 

69.65 

25.01 

74 

75 

70.91 

24.42 

70.81 

24.73 

70.70 

25.04 

70.59 

25.34 

75 

76 

71.86 

24.74 

71.75 

25.06 

71.64 

25.37 

71.53 

25.68 

76 

77 

72.80 

25.07 

72.69 

25.39 

72.58 

25.70 

72.47 

26.02 

77 

78 

73.75 

25  .  39 

73.64 

25.72 

73.53 

26.04 

73.41 

26.36 

78 

79 

74.70  J25.72 

74.58 

26.05 

74.47 

26.37 

74.35 

26.70 

79 

80 

75.  64  1  26.  05 

75.53 

26.38 

75.41 

26.70 

75.29 

27.03 

80 

81 

73.59 

26.37 

76.47 

26.70 

76.35 

27.04 

76.24 

27.37 

81 

82 

77.53 

26.70 

77.42 

27.03 

77.30 

27.37 

77.18 

27.71 

82 

83 

78.48 

27.02 

78.36 

27.36 

78.24 

27.71 

78.12 

28.05 

83 

84 

79.42 

27.35 

79.30 

27.69 

79.18 

28.04 

79.06 

28.39 

84 

85 

80.37 

27.67 

80.25 

28.02 

80.12 

28.37 

80.00 

28.72 

85 

86 

81.31 

28.00 

81.19 

28.35 

81.07 

28.71 

80.94 

29.06 

86 

87 

82.26 

28.32 

82.14 

28.68 

82.01 

29.04 

81.88 

29.40 

87 

88 

83.21 

28.65 

83.08 

29.01 

92.95 

29.37 

82.82 

29.74 

88 

89 

84.15 

28.98 

84.02 

29.34 

83.90 

29.71 

|83.76 

30.07 

89 

90 

85.10 

29.30 

84.97 

29.67 

84.84 

30.04 

84.71 

30.41 

90 

91 

86.04 

29.63 

85.91 

30.00 

85.78 

30.38 

85.65 

30.75 

91 

92 

86.99 

29.95 

86.86 

30.33 

86.72 

30.71 

86.59 

31.09 

92 

93 

87.93 

30.28 

87.80 

30.66 

87.67 

31.04 

87.53 

31.43 

93 

94 

88.88 

30.60 

88.74 

30.99 

88.61 

31.38 

88.47 

31.76 

94 

95 

89.82 

«0.93 

89.69 

31.32 

89.55 

31.71 

89.41 

32.10 

95 

96 

90.77 

31.25 

90.63 

31.65 

90.49 

32.05 

90.35 

32.44 

96 

97 

91.72 

31.58 

91.58 

31.98 

91.44 

32.38 

91.29 

32.78 

97 

93 

92.66 

31.91 

92.52 

32.31 

92.38 

32.71 

92.24 

33.12 

98 

99 

93.61 

32.23 

93.46 

32.64 

93.32 

33.05 

93.18 

33.45 

99 

100 

94.55 

32.56 

94.41 

32.97 

94.26 

33.38 

94.12 

33.79 

100 

§ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

u 

c 

1 

71  Deg. 

70|  Deg. 

701  Deg. 

70J  Deg. 

"OT 

S 

i 

TRAVERSE    TABLE. 


i 

20  Deg. 

204  Deg. 

20|  Deg. 

20|  Deg. 

O 

s 
P 

itance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.94 

0.34 

0.94 

0.35 

0.94 

0.35 

0.94 

0.35 

1 

2 

1.88 

0.68 

1.88 

0.69 

1.87 

0.70 

1.87 

0.71 

2 

3 

2.82 

1.03 

2.81 

1.04 

2.81 

1.05 

2.81 

1.06 

3 

4 

3.76 

1.37 

3.75 

1.38 

3.75 

1.40 

3.74 

1.42 

4 

5 

4.70 

1.71 

4.69 

1.73 

4.68 

1.75 

4.  08 

1.77 

5 

6 

5.64 

2.05 

5.63 

2.08 

5.62 

2.10 

5.61 

2.13 

6 

7 

6.58 

2.39 

6.57 

2.42 

6.56 

2.45 

6.55 

2.48 

7 

8 

7.52 

2.74 

7.51 

2.77 

7.49 

2.80 

7.48 

2.83 

8 

9 

8.46 

3.08 

8.44 

3.12 

8.43 

3.15 

8.42 

3.19 

9 

10 

9.40 

3.42 

9.38 

3.46 

9.37 

3.50 

9.35 

3.54 

10 

11 

10.34 

3.76 

10.32 

3.81 

10.30 

3.85 

10.29 

3.90 

11 

12 

11.28 

4.10 

11.26 

4.15 

11.24 

4.20 

11.22 

4.25 

12 

13 

12.22 

4.45 

12.20 

4.50 

12.18 

4.55 

12.16 

4.61 

13 

14 

13.16 

4.79 

13.13 

4.85 

13.11 

4.90 

13.09 

4.96 

14 

15 

14.10 

5.13 

14.07 

5.19 

14.05 

5.25 

14.03 

5.31 

15 

16 

15.04 

5.47 

15.01 

5.54 

14.99 

5.60 

14.96 

5.67 

16 

17 

15.97 

5.81 

15.95 

5.88 

15.92 

5.95 

15.90 

6.02 

17 

18 

16.91 

6.16 

16.89 

6.23 

16.86 

6.30 

16.83 

6.38 

18 

19 

17.85 

6.50 

17.83 

6.58 

17.80 

6.65 

17.77 

6.73 

19 

20 

18.79 

6.84 

18.76 

6.92 

18.73 

7.00 

18.70 

7.09 

20 

21 

19.73 

7.18 

19.70 

7.27 

19.67 

7.35 

19.64 

7.44 

21 

22 

20.67 

7.52 

20.64 

7.61 

20.61 

7.70 

20.57 

7.79 

22 

23 

21.01 

7.87 

21.58 

7.96 

21.54 

8.05 

21.51 

8.15 

23 

24 

22.55 

8.21 

22.52 

8.31 

22.48 

8.40 

22.44 

8.50 

24 

25 

23.49 

8.55 

23.45 

8.65 

23.42 

8.76 

23.38 

8.86 

25 

26 

24.43 

8.89 

24.39 

9.00 

24.35 

9.11 

24.31 

9.21 

26 

27 

25.37 

9.23 

25.33 

9.35 

25.29 

9.46 

25.25 

9.57 

27 

28 

26.31 

9.58 

26.27 

9.69 

26.23 

9.81 

26.18 

9.92 

28 

29 

27.25 

9.92 

27.21 

10.04 

27.16 

10.16 

27.12 

10.27 

29 

30 

28.191  10.26 

28.15 

10.38 

28.10 

10.51 

28.05 

10.63 

30 

31 

29.13 

10.60 

29.08 

10.73 

29.04 

10.86 

28.99 

10.98 

31 

32 

30.07 

10.94 

30.02 

11.08 

29.97 

11.21 

29.92 

11.34 

32 

33 

31.01 

11.29 

30.96 

11.42 

30.91 

11.56 

30.86 

11.69 

33 

34 

31.95 

11.63 

31.90 

11.77 

31.85 

11.91 

31.79 

12.05 

34 

35 

32.89 

11.97 

32.84 

12.11 

32.78 

12.26 

32.73 

12.40 

35 

36 

33.83 

12.31 

33.77 

12.46 

33.72 

12.61 

33.66 

12.75 

36 

37 

34.77 

12.65 

34.71 

12.81 

34.66 

12.96 

34.60 

13.11 

37 

38 

35.71 

13.00 

35.65 

13.15 

35.59 

13.31 

35.54 

13.46 

38 

39 

36.65 

13.34 

36  .  59 

13.50 

30.53 

13.66 

36.47 

13.82 

39 

40 

37.59 

13.68 

37.53 

13.84 

37.47 

14.01 

37.41 

14.17 

40 

41 

38.53 

14.02 

38.47 

14.19 

38.40 

14.36 

38.34 

14.53 

41 

42 

39.47 

14.36 

39.40 

14.54 

39.34 

14.71 

39.28 

14.88 

42 

43 

40.41 

14.71 

40.34 

14.88 

40.28 

15.06 

40.21 

15.23 

43 

44 

41.35 

15.05 

41  .28 

15.23 

41.21 

15.41 

41.15 

15.59 

44 

45 

42.29 

15.39 

42.22 

15.58 

42.15 

15.76 

42.0» 

15.94 

45 

46 

43.23 

15.73 

43.16 

15.92 

43.09 

16.11 

43.02 

16.30 

46 

47 

44.17 

16.07 

44.09 

16.27 

44.02 

16.46 

43.95 

16.65 

47 

48 

45.11 

16.42 

45.03 

16.61 

44.96 

16.81 

44.89 

17.01 

48 

49 

46.04 

16.76 

45.97 

16.96 

45.90 

17.16 

45.82 

17.36 

49 

50 

46.98 

17.10 

46.91 

17.31 

46.83 

17.51 

46.76 

17.71 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

O) 

§ 

J 

s 

70  Deg. 

69|  Deg. 

69$  Deg. 

69i  Deg. 

d 

73 

3 

TBAVERSE    TABLE. 


d 

20  Deg. 

204  Deg. 

20£  Deg. 

20|  Deg. 

c. 

p 

• 

S- 

3 

n 

0 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

47.92 

17.44| 

r.ss 

17.65 

47.77 

17.86! 

47.69 

18.07 

51 

52 

43.86 

17.79 

48.79 

18.00 

48.71 

18.21 

48.63 

18.42 

52 

53 

49.80 

18.13; 

49.72 

18.34 

49.64 

18.56 

49.56 

18.78 

53 

54 

50.74 

18.47 

50.66 

18.69 

50.58 

18.91 

50.50 

19.13 

54 

55 

51.68 

18.81 

51.60 

19.04 

51.52 

19.26! 

51.43 

19.49 

55 

56 

52.62 

19.15 

52.54 

19.38 

52.45 

19.611 

52.37 

19.84 

56 

57 

53.56 

19.50  i 

53.48 

19.73 

53.39 

19.96; 

53.30 

20.19 

57 

58 

54.50 

19.84 

54.42 

20.07 

54.33 

20.31 

54.24 

20.55 

58 

59 

55.44 

20.13  ! 

55.35 

20.42 

55.26 

20.66; 

55.17 

20.90 

59 

60 

56.38 

20.52 

56.29 

20.77 

56.20 

21.01 

56.11 

21.26 

60 

61 

57.32 

20.86 

57.23 

21.11 

57.14 

21.36 

57.04 

21.61 

61 

62 

58.26 

21.21 

58.17 

21.46 

58.07 

21.71 

57.98 

21.97 

62 

63 

59.20 

21.55 

59.11 

21.81 

59.01 

22.06 

58.91 

22.32 

63 

64 

60.14 

21.89 

60.04 

22.15 

59.95 

22.41 

59.85 

22.67 

64 

65 

61.08 

22.23  1 

60.93 

22.50 

60.88 

22.76 

60.73 

23.03 

65 

66 

62.02 

22.57 

61.92 

22.84 

61.82 

23.11 

61.72 

23.38 

66 

67 

62.96 

22.92 

62.86 

23.19 

62.76 

23.46 

62.65 

23.74 

67 

68 

63.90 

23.26 

63.80 

23.54 

63.69 

23.81 

63.59 

24.09 

68 

69 

64.84 

23.60 

64.74 

23.88 

64.63 

24.16 

64.52 

24.45 

69 

70 

65  .  78 

23.94 

65.67 

24.23 

65.57 

24.51 

65.46 

24.80 

70 

71 

66.72 

24.28 

66.61 

24.57 

66.50 

24.86 

66.39 

25.15 

71 

72 

67.66    24.63 

67.55 

24.92 

67.44 

25.21 

67.33 

25.51 

72 

73 

68.60 

24.97 

68.49 

25.27 

68.38 

25.57 

68.26 

25.86 

73 

74 

69.54 

25.31 

69.43 

25.61 

69.31 

25.92 

69.20 

26.22 

74 

75 

70.48 

25.65 

70.36 

25.96 

70.25 

26.27 

70.14 

26.57 

75 

76 

71.42 

25.99 

71.30 

26.30 

71.19 

26.62 

71.07 

26.93 

76 

77 

72.36 

26.34 

72.24 

26.65 

72.12 

26.97 

72.01 

27.28 

77 

78 

73.30 

26.68 

73.18 

27.00 

73.06 

27.32 

72.94 

27.63 

78 

79 

74.24 

27.02 

74.12 

27.34 

74.00 

27.67 

73.88 

27.99 

79 

80 

75.18 

27.36: 

75.06 

27.69 

74.93 

28.02 

74.81 

28.34 

80 

81 

76.12 

27.70: 

75.99 

28.04 

75.87 

28.37 

75.75 

28.70 

81 

82 

77.05 

28.05| 

76.93 

23.38 

76.81 

28.72 

76.68 

29.05 

82 

83 

77.99 

28.39 

77.87 

28.73 

77.74 

29.07 

77.62 

29.41 

83 

84 

78.93 

28  .  73  ! 

78.81 

29.07 

78.68 

29.42 

78.55 

29.76 

84 

85 

79.87 

29.07! 

79.75 

29.42 

79.62 

29.77 

79.49 

30.11 

85 

86 

80.81 

29.41  i 

80.68 

29.77 

80.55 

30.12 

'80.42 

30.47 

86 

87 

81.75 

29.76^ 

81.62 

30.11 

81.49 

30.47 

81.36 

30.82 

87 

88 

82.69 

30.10; 

82.56 

30.46 

82.43 

30.82 

182.29 

31.18 

88 

89 

83.63 

30.44! 

83.50 

30.80 

83.36 

31.17 

183.23 

31.53 

89 

90    84.57 

30.78     84.44    31.15 

84.30 

31.52 

i84.!6 

31.89 

90 

91 

85.51 

31.12 

85.38 

31.50 

85.24 

31.87 

J85.10 

32.24 

91 

92 

86.45 

31.47 

86.31 

31.84* 

86.17 

32.22 

86.03 

32.59 

92 

93 

87.39 

31.81 

87.25 

32.19 

87.11 

32.57 

86.97 

32.90 

93 

94 

88.33 

32.15 

88.19 

32.54 

88.05 

32.92 

87.90 

33.30 

94 

95 

89.27 

32.49 

89.13 

32.88 

88.98 

33.27 

88.84 

33.66 

95 

96 

90.21 

32.83 

90.07 

33.23 

89.92 

33.62 

89.77 

34.01 

96 

97    91.15 

33.18 

91.00 

33.57 

90.86 

33.97 

90.71 

34.37 

97 

98    92.09 

33.52 

91.94 

33.92 

91.79 

34.32 

91.64 

34.72 

98 

99 

93.03 

33.86 

92.88 

34.27 

92.73 

34.67 

92.58 

35.07 

99 

100 

93.97 

34.20 

93.82 

34.61 

93.67 

35.02 

93.51 

35.43 

100 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

33 

•£ 

70  Deg. 

89|  Deg. 

69£  Deg. 

69i  Deg. 

5 

44 


TRAVERSE    TABLE. 


o 

5' 

21  Deg. 

21^  Deg. 

•* 

21  1  Deg. 

21|  Deg. 

C 

en" 

3 
(5 
ffl 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 
O 

a 

1 

0.93 

0.36 

0.93 

0.36 

0.93 

O.oT 

0.93 

0.37 

F 

2 

1.87 

0.72 

1.86 

0.72 

'  1.86 

0.73 

1.86 

0.74 

2 

3 

2.80 

1.08 

2.80 

1.09 

2.79 

1.10 

2.79 

1.11 

3 

4 

3.73 

1.43 

3.73 

1.45 

3.72 

1.47 

3.72 

1.48 

4 

5 

4.67 

1.79 

4.66 

1.81 

4.65 

1.83 

4.64 

1.85 

5 

6 

5.60 

2.15 

5.59 

2.17 

5.58 

2.20 

5.57 

2.22 

6 

7 

6.54 

2.51 

6.52 

2.54 

6.51 

2.57 

6.50 

2.59 

7 

8 

7.47 

2.87 

7.46 

2.90 

7.44 

2.93 

7.43 

2.96 

8 

9 

8.40 

3.23 

8.39 

3.26 

8.37 

3.30 

8.36 

3.34 

9 

10 

9.34 

3.58 

9.32 

3.62 

9.30 

3.67 

9.29 

3.71 

10 

11 

10.27 

3.94 

10.25 

3.99 

10.23 

4.03 

10.22 

4.08 

11 

12 

11.20 

4.30 

11.18 

4.35 

11.17 

4.40 

11.15 

4.45 

12 

13 

12.14 

4.66 

12.12 

4.71 

12.10 

4.76 

12.07 

4.82 

13 

14 

13.07 

5.02 

13.05 

5.07 

13.03 

5.13 

13.00 

5.19 

14 

15 

14.00 

5.39 

13.98 

5.44 

13.96 

5.50 

13.93 

5.56 

15 

16 

14.94 

5.73 

14.91 

5.80 

14.89 

5.86 

14.86 

5.93 

16 

17 

15.87 

6.09 

15.84 

6.16 

15.82 

6.23 

15.79 

6.30 

17 

18 

16.80 

6.45 

16.78 

6.52 

16.75 

6.60 

16.72 

6.67 

18 

19 

17.74 

6.81 

17.71 

6.89 

17.68 

6.96 

17.65 

7.04 

19 

20 

18.67 

7.17 

18.64 

7.25 

18.61 

7.33 

18.58 

7.41 

20 

21 

19.61 

7.53 

19.57 

7.61 

19.54 

7.70 

19.50 

7.78 

21 

22 

20.54 

7.88 

20.50 

7.97 

20.47 

8.06 

20.43 

8.15 

22 

23 

21.47 

8.24 

21.44 

8.34 

21.40 

8.43 

21.36 

8.52 

23 

24 

22.41 

8.60 

22.37 

8.70 

22.33 

8.80 

22.29 

8.89 

24 

25 

23.34 

8.96 

23.30 

9.06 

23.26 

9.16 

23.22 

9.26 

25 

26 

24.27 

9.32 

24.23 

9.42 

24.19 

9.53 

24.15 

9.63 

26 

25.21 

9.68 

25.16 

9.79 

25.12 

9.90 

25.08 

10.01 

27 

28 

26.14 

10.03 

26.10 

10.15 

26.05 

10.26 

26.01 

10.38 

28 

29 

27.07 

10.39 

27.03 

10.51 

26.98 

10.63 

26.94 

10.75 

29 

30 

28.01 

10.75 

27.96 

10.87 

27.91 

11.00 

27.86 

1U12 

30 

31 

28.94 

11.11 

28.89 

11.24 

28.84 

11.36 

28.79 

11.49 

31 

32 

29.87 

11.47 

29.82 

11.60 

29.77 

11.73 

29.72 

11.86 

32 

33 

30.81 

11.83 

30.76 

11.96 

30.70 

12.09 

30.65 

12.23 

33 

34 

31.74 

12.18 

31.69 

12.32 

31.63 

12.46 

31.58 

12.60 

34 

35 

32.68 

12.54 

32.62 

12.69 

32.56 

12.83 

32.51 

12.97 

35 

36 

33.61 

12.90 

33.55 

13.05 

33.50 

13.19 

33.44 

13.34 

36 

37 

34.54 

13.26 

34.48 

13.41 

34.43 

13.56 

.34.37 

13.71 

37 

38 

35.48 

13.62 

35.42 

13.77 

35.36 

13.93 

35.29 

14.08 

38 

39 

'36.41 

J3.98 

36.35 

14.14 

36.29 

14.29 

36.22 

14.45 

39 

40 

37.34 

14.33 

37.28 

14.50 

37.22 

14.66 

37.15 

14.82 

40 

41 

38.28 

14.69 

38.21 

14.86 

38.15 

15.03 

38.08 

15.19 

41 

42 

39.21 

15.05 

39.14 

15.22 

39.08 

15.39 

39.01 

15.56 

42 

43 

40.14 

15.41 

40.08 

15.58 

40.01 

15.76 

39.94 

15.93 

43 

44 

41.08 

15.77 

41.01 

15.95 

40.94 

16.13 

40.87 

16.30 

44 

45 

42.01 

16.13 

41.94 

16.31 

41.87 

16.49 

41.80 

16.68 

45 

46 

42.94 

16.48 

42.87 

16.67 

42.80 

16.86 

42.73 

17.05 

46 

47 

43.88 

16.84 

43.80 

17.03 

43.73 

17.23 

43.65 

17.42 

47 

48 

44.81 

17.20 

44.74 

17.40 

44.66 

17.59 

44.58 

17.79 

48 

49 

45.75 

17.56 

45.67 

17.76 

45.59 

17.96 

45.51 

18.16 

49 

50 

46.68 

17.92 

46.60 

18.12 

46.52 

18.33 

46.44 

18.53 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

,  Lat. 

Dep. 

Lat. 

i 

.a 

Q 

69  Deg. 

68|  Deg. 

681  Deg. 

684  Deg. 

.2 

Q 

| 

TRAVERSE    TABLE. 


45 


c 

21  Deg. 

21}  Deg. 

21i  Deg. 

21  1  Deg. 

O 

55° 

? 

5" 

9 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

n 

9 

51 

47.61 

18.28 

47.53 

18.48 

47.45 

18.69 

47.37 

18.90 

51 

52 

48.55 

18.64 

48.46 

18.85 

48.38 

19.06 

48.30 

19.27 

52 

53 

49.48 

18.99 

49.40 

19,21 

49.31 

19.42 

49.23 

19.64 

53 

54 

50.41 

19.35 

50.33 

19.57 

50.24 

19.79 

50.16 

20.01 

54 

55 

51.35 

19.71 

51.26 

19.93 

51.17 

20.16 

51.08 

20.38 

55 

56 

52  28 

20.07 

52.19 

20.30 

52.10 

20.52 

52.01 

20.75 

56 

57 

53  21 

20.43 

53.12 

20.66 

53.03 

20.89 

52.94 

21.12 

57 

58 

54.15 

20.79 

54.06 

21.02 

53.96 

21.26 

53.87 

21.49 

58 

59 

55.08 

21.14 

54.99 

21.38 

54.89 

21.62 

54.80 

21.86 

59 

60 

56.01 

21.50 

55.92 

21.75 

55.83 

21.99 

55.73 

22.23 

60 

61 

56  .  95 

21.86 

56.85 

22.11 

56.76 

22.36 

56.66 

22.60 

61 

62 

57.88 

22.22 

57.78 

22.47 

57.69 

22.72 

57.59 

22.97 

62 

63 

58.82 

22.58 

58.72 

22.83 

58.62 

23.09 

58.52 

23.35 

63 

64 

59.75 

22.94 

59.65 

23.20 

59.55 

23.46 

59.44 

23.72 

64 

65 

60.68 

23.29 

60.58 

23.56 

60.48 

23.82 

60.37 

24.09 

65 

66 

61.62 

23.65 

61:.  51 

23.92 

61.41 

24.19 

61.30 

24.46 

66 

67 

62.55 

24.01 

62.44 

24.28 

62.34 

24.56 

62.23 

24.83 

67 

68 

63.48 

24.37 

63.38 

24.65 

63.27 

24.92 

63.16 

25.20 

68 

69 

64.42 

24.73 

64.31 

25.01 

64.20 

25.29 

64.09 

25.57 

69 

70 

65.35 

25.09 

65.24 

25.37 

65.13 

25.66 

65.02 

25.94 

70 

71 

66.38 

25.44 

66.17 

25.73 

66.06 

26.02 

65.95 

26.31 

71 

72 

67.22 

25.80 

67.10 

26.10 

66.99 

26.39 

66.87 

26.68 

72 

73 

68.15 

26.16 

68.04 

26.46 

67.92 

26.75 

67.80 

27.05 

73 

74 

69.08 

26.5211  68.97 

26.82 

68.85 

27.12 

68.73 

27.42 

74 

75 

70.02 

26.88  !  69.90 

27.18 

69.78 

27.49 

69.66 

27.79 

75 

76 

70.95 

27.24     70.83 

27.55 

70.71 

27.85 

70.59 

28.16 

76 

77 

71.89 

27.59 

71.76 

27.91 

71.64 

28.22 

71.52 

28.53 

77 

78 

72.82 

27.95 

72.70 

28.27 

72.57 

28.59 

72.45 

28.90 

78 

79 

73.75 

28.31 

73.63 

28.63 

73.50 

28.95 

73.38 

29.27 

7y 

80 

74.69 

28.67 

74.56 

29.00 

74.43 

29.32 

74.30 

29.64 

80 

81 

75.62 

29.03 

75.49 

29.36 

75.36 

29.69 

75.23 

30.02 

81 

82 

76.55 

29.39 

76.42 

29.72 

76.29 

30.05 

76.16 

30.39 

82 

83 

77.49 

29.74 

77.36 

30.08 

77.22 

30.42 

77.09 

30.76 

83 

84 

78.42 

30.10 

78.29 

30.44 

78.16 

30.79 

78.02 

31.13 

84 

85 

79.35 

30.46 

79.22 

30.81 

79.09 

31.15 

78.95 

31.50 

85 

86 

80.29 

30.82 

80.15 

31.17 

80.02 

31.52 

79.88 

31.87 

86 

87 

81.22 

31.18 

81.08 

31.53 

80.95 

31.89 

80.81 

32'.  24 

87 

88 

82.16 

31.54 

82.02 

31.89 

81.88 

32.25 

81.74 

32.61 

88 

89 

83.09 

31.89 

82.95 

32.26 

82.81 

32.62 

82.66 

32.98 

39 

90 

84.02 

32.25 

83.88 

32.62 

83.74 

32.99 

83.59 

33.35 

90 

91 

84.96 

32.61 

84.81 

32.98 

84.  6T 

33.35 

84.52 

33.72 

91 

92 

85.89 

32.97 

85.74 

33.34 

85.60 

33.72 

85.45 

34.09 

92 

93 

86.  82 

33.33 

86.68 

33.71 

86.53 

34.08 

86.38 

34.46 

93 

94 

87.76 

33.69 

87.61 

34.07 

87.46 

34.45 

87.31 

34.83 

94 

95 

88.69 

34.04 

88.54 

34.43 

88.39 

34.82 

88.24 

35.20 

95 

96 

89.62 

34.40 

89.47 

34.79 

89.32 

35.18 

89.17 

35.57 

96 

97 

90.56 

34.76 

90.40 

35.16 

90.25 

35.55 

90.09 

35.94 

97 

98 

91.49 

35.12 

91.34 

35.52 

91.18 

35.92 

91.02 

36.31 

98 

99 

92.42 

35.48 

92.27 

35.88 

92.11 

36.28 

91.95 

36.69 

99 

100 

93.36 

35.84 

93.20 

36.24     93.04 

36.65 

92.88 

37.06 

100 

I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o' 
g 

1 
fl 

«9  Deg. 

68.|  Deg. 

68£  Deg. 

68i  Deg. 

cd 

In 

S 

40 


TRAVERSE    TABLE. 


o 

QQ" 

22  Deg. 

22|  Deg. 

22^  Deg. 

22|  Deg. 

B 
I' 

1 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

a 
P 

1 

0.93 

0.37 

0.93 

0.38 

0.92 

0.38 

0.92 

0.39 

1 

2 

1.85 

0.75 

1.85 

0.76 

1.85 

0.77 

1.84 

0.77 

2 

i     3 

2.78 

1.12 

2.78 

1.14 

2.77 

1.15 

2.77 

1.16 

3 

4 

3.71 

1.50 

3.70 

1.51 

3.70 

1.53 

3.69 

1.55 

4 

;    5 

4.64 

1.87 

4.63 

1.89 

4.62 

1.91 

4.61 

1.93 

5 

i      6 

5.56 

2.25 

5.55 

2.27 

5.54 

2.30 

5.53 

2.32 

6 

i      7 

6.49 

2.62 

6.48 

2.65 

6.4V 

2.68 

6.46 

2.71 

7 

8 

7.42 

3.00 

7.40 

3.03 

7.39 

3.06 

7.38 

3.09 

8 

9 

8.34 

3.37 

8.33 

3.41 

8.31 

3.44 

8.30 

3.48 

9 

10 

9.27 

3.75 

9.26 

3.79 

9.24 

3.83 

9.22 

3.87 

10 

•    11 

10.20 

4.12 

10.18 

4.17 

10.16 

4.21 

10.14 

4.25 

11 

:    12 

11.13 

4.50 

11.11 

4.54 

11.09 

4.59 

11.07 

4.64 

12 

13 

12.05 

4.87 

12.03 

4.92 

12.01 

4.97 

11.99 

5.03 

13 

14 

12.98 

5.24 

12.96 

5.30 

12.33 

5.36 

12.91 

5.41 

14 

15 

13.91 

5.62 

13.88 

5.68 

13.86 

5.74 

13.83 

5.80 

15 

i  16 

14.83 

5,99 

14.81 

6.06 

14.78 

6.12 

14.76 

6.19 

16 

'    17 

15.76 

6.37 

15.73 

6.44 

15.71 

6.51 

15.68 

6.57 

17 

:  is 

16.69 

6.74 

16.66 

6.82 

16.63 

6.89 

16.60 

6.96 

18 

19 

17.62 

7.12 

17.59 

7.19 

17.55 

7.27 

17.52 

7.35 

19 

20 

18.54 

7.49 

18.51 

7.57 

18.48 

7.65 

18.44 

7.73 

20 

21 

19.47 

7.87 

19.44 

7.95 

19.40 

8.04 

19.37 

8.12 

21 

'    22 

20.40 

8.24 

20.36 

8.33 

20.33 

8.42 

20.29 

8.51 

22 

1    23 

21.33 

8.62 

21.29 

8.71 

21.25 

8.80 

21.21 

8.89 

23 

24 

22.25 

8.99 

22.21 

9.09 

22.17 

9.18 

22.13 

9.28 

24 

25 

23.18 

9.37 

23.14 

9.47 

23.10 

9.57 

23.05 

9.67 

25 

26 

24.11 

9.74 

24.06 

9.84 

24.02 

9.95 

23.98 

10.05 

26 

,   27 

25.03 

10.11 

24.99 

10.22 

24.94 

10.33 

24.90 

10.44 

27 

28 

25  .  96 

10.49 

25.92 

10.60 

25.87 

10.72 

25.82 

10.83 

28 

»29 

26.89 

10.86 

26.84 

10.98 

26.79 

11.10 

26.74 

11.21 

29 

30 

27.82 

11.24 

27.77 

11.36 

27.72 

11.48 

27.67 

11.60 

30 

31 

28.74 

11.61 

28.69 

11.74 

28.64 

11.86 

28.59 

11.99 

31 

32 

29.67 

11.99 

29.62 

12.12 

29.56 

12.25 

29.51 

12.37 

32 

33 

30.60 

12.36 

30.54 

12.50 

30.49 

12.63 

30.43 

12.76 

33 

34 

31.52 

12.74 

31.47 

12.87 

31.41 

13.01 

31.35 

13.15 

34 

35 

32.45 

13.11 

32.39 

13.25 

32.34 

13.39 

32.28 

13.53 

35 

36 

33.38 

13.49 

33.32 

13.63 

33.26 

13.78 

33.20 

13.92 

36 

•    37 

34.31 

13.86 

34.24 

14.01 

34.18 

14.16 

34.12 

14.31 

37 

38 

35.23 

14.24 

35.17 

14.39 

35.11 

14.54 

35.04 

14.70 

38 

39 

36.16 

14.61 

36.10 

14.77 

36.03 

14.92 

35.97 

15.08 

39 

40 

37.09 

14.98 

37.02 

15.15 

36.96 

15.31 

36.89 

15.47 

40 

41 

38.01 

15.36 

37  .  95 

15.52 

37.88 

15.69 

37.81 

15.86 

41 

42 

38.94 

15.73 

38.87 

15.90 

38.80 

16.07 

38.73 

16.24 

42 

43 

39.87 

16.11 

39.80 

16.28 

39.73 

16.46 

39.65 

16.63 

43 

44 

40,.  80 

16.48 

40.72 

16.66 

40.65 

16.84 

40.58 

17.02 

44 

45 

41.72 

16.86 

41.65 

17.04 

41.57 

17.22 

41.50 

17.40 

45 

46 

42.65 

17.23 

42.57 

17.42 

42.50 

17.60 

42.42 

17.79 

46 

47 

43.58 

17.61 

43.50 

17.80 

43.42 

17.99 

43.34 

18.18 

47 

48 

44.50 

17.98 

44.43 

18.18 

44.35 

18.37 

44.27 

18.56 

48 

49 

45.43 

18.36 

45'.  35 

18.55 

45.27 

18.75 

45.19 

18.95 

49 

;     50 

46.36 

18.73 

46.28 

18.93 

46.19 

19.13 

46.11 

19.34 

50 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

o 

a 

':      .3 

68  Deg. 

67|  Deg. 

67A  Deg. 

67i  Deg. 

• 

3 

Til  A  VERSE    TABLE. 


47 


c 

So* 

22  Deg. 

22*  Deg. 

22£  Deg. 

22|  Deg. 

O 
5' 

p 

? 

3 

n 
a 

Lat.  |  Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

51 

47.29 

19.10 

47.20 

19.31 

47.12 

19.52 

47.03 

19.72 

51 

52 

4S.21 

19.48 

48.13 

19;69 

48.04 

19.90 

47.95 

20.11 

52 

53 

49.14 

19.85 

49.05 

20.07 

48.97 

20.28 

48.88 

20.50 

53 

54 

50.07 

20.23 

49.98 

20.45 

49.89 

20.66 

49.80 

20.88 

54 

55 

51.00 

20.60 

50.90 

20.83 

50.81 

21.05 

50.72 

21.27 

55 

56 

51.92 

20.98 

51.83 

21.20 

51.74 

21.43 

51.64 

21.66 

56 

57 

52.85 

21.35, 

52.76 

21.58 

52.66 

21.81 

52.57 

22.04 

57 

58 

53.78 

21.73 

53.68 

21.96 

53.59 

22.20 

53.49 

22.43 

58 

59 

54.70 

22.10 

54.61 

22.34 

54.51 

22.58 

54.41 

22.82 

59 

60 

55.63 

22.48 

55.53 

22.72 

55.43 

22.96 

55.33 

23.20 

60 

61 

56  .  56 

22.85 

56.47 

23.10 

56.36 

23.34 

56.25 

23.59 

61 

62 

57.49 

23.23 

57.38 

23.48 

57.28 

23.73 

57.18 

23.98 

62 

63 

58.41 

23.60 

58.31 

23.85 

58.20 

24.11 

58.10 

24.38 

63 

64 

59.34 

23.97 

59.23 

.24.23 

59.13 

24.49 

59.02 

24.75 

64 

65 

60.27 

24.35 

60.16 

24.61 

60.05 

24.87 

59.94 

25.14 

65 

66 

61.19 

24.72 

61.09 

24.99 

60.98 

25.26 

60.87 

25.52 

66 

67 

62.12 

25.10 

62.01 

25.37 

61.90 

25.64 

61.79 

25.91 

67 

68 

63.05 

25.47 

62.94 

25.75 

62.82 

26.02 

62.71 

26.30 

68 

69 

63.98 

25.85 

63.86 

26.13 

63.75 

26.41 

63.63 

26.68 

69 

70 

64.90 

26.22 

64.79 

26.51 

64.67 

26.79 

64.55 

27.07 

70 

71 

65.83 

26.60 

65.7! 

26.88 

65.60 

27.17 

65.48 

27.46 

71 

72 

66.76 

26.97 

66.64 

27.26 

66.52 

27.55 

66.40 

27.84 

72 

73 

67.68 

27.35 

67.56 

27.64 

67.44 

27.94 

67.32 

28.23 

73 

74 

68.61 

27.72 

68.49 

28.02 

68.37 

28.32 

68.24 

28.62 

74 

75 

69.54 

28.10 

69.42 

28.40 

69.29 

28.70 

69.17 

29.00 

75 

76 

70.47 

23.47 

70.34 

28.78 

70.2JL 

29.08 

70.09 

29.39 

76 

77 

71.39 

28.84 

71.27 

29.16 

71.  R 

29.47 

71.01 

29.78 

77 

78 

72.32 

29.22 

72.19 

29.53 

72.06 

29.85 

71.93 

30.16 

78 

79 

73.25 

29.59 

73.12 

29.91 

72.99 

30.23 

72.85 

30.55 

79 

80 

74.17 

29.97 

74.04 

30.29 

73.91 

30.61 

73.78 

30.94 

80 

81 

75.10 

30.34 

74.97 

30.67 

74.83 

31.00 

74.70 

31.32 

81 

82 

76.03 

30.72 

75.89 

31.05 

75.76 

31.38 

75.62 

31.71 

82 

83 

76.96 

31.09 

76.82 

31.43 

76.68 

31.76 

76.54 

32.10 

83 

84 

77.88 

31.47 

77.75 

31.81 

77.61 

32.15 

77.46 

32.48 

84 

85 

78.81 

31.84 

78.67 

32.19 

78.53 

32.53 

78.39 

32.87      85 

86 

79.74 

32.22 

79.60 

32.56 

79.45 

32.91 

79.31 

33.26      86 

87 

80.66 

32.59 

80.52 

32.94 

80.38 

33.29 

80.23 

33.64 

87 

83 

81.59 

32.97 

81.45 

33.32 

81.30 

33.68 

81.15 

34.03 

88 

89 

82.52 

33.34 

82.37 

33.70 

82.23 

34.06 

82.08 

34.42 

89 

90 

83.45 

33.71 

83.30 

34.08 

83.15 

34.44 

83.00 

34.80 

90 

91 

84.37 

34.09 

84'.22 

34.46 

84.07 

34.82 

83.92 

35.19 

91 

92 

85.30 

34.46 

85.15 

34.84 

85.00 

35.21 

84.84 

35.58 

92 

93    86.23 

34.84 

86.08 

35.21 

85.92 

35.59 

85.76 

35.96 

93 

94    87.16 

35.21 

87.00    35.59 

86.84 

35.97 

86.69 

36.35 

94 

95    88.08 

35.59 

87.93135.97 

87.77 

36.35 

87.61 

36.74 

95 

96    89.01 

35.96 

88.85    36.35 

88.69 

36.74 

88.53 

37.12 

96 

97189.94 

36.34 

89.78;  36.73 

80.62 

37.12 

8».45 

37.51 

97 

98 

90.86 

36.71 

90.70    37.11 

90.54 

37.50 

90.38 

37.90 

98 

99 

91.79 

37.09 

91.63    37.49 

91.46 

37.89 

91.30 

38.28 

99 

100 

92.72 

37.46 

92.55!  37.86 

92.39 

38.27 

92.22 

38.67 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

a 

« 

£ 

X 

3 

68  Deg. 

67|  Deg. 

67£  Deg. 

-  67i  Deg. 

Q 

TRAVERSE    TABLE. 


O 

r 

23  Deg. 

23J  Deg. 

23|  Deg. 

23|  Deg. 

i 

1 

1 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0.92 

0.39 

0.92 

0.39 

0.92 

0.40 

0.92 

0.40 

i 

2 

1.84 

0.78 

1.84 

0.79 

1.83 

0.80 

1.83 

0.81 

2 

3 

2.76 

1.17 

2.76 

1.18 

2.75 

1.20 

2.75 

1.21 

3 

4 

3.68 

1.56 

3.68 

1.58 

3.67 

1.59 

3.66 

1.61 

4 

5 

4.60 

1.95 

4.59 

1.97 

4.59 

1.99 

4.58 

2.01 

5 

6 

5.52 

2.34 

5.51 

2.37 

5.50 

2.39 

5.49 

2.42 

6 

7 

6.  '14 

2.74 

6.43 

2.76 

6.42 

2.79 

6.41 

2.82 

7 

8 

7.36 

3.13 

7.35 

3.16 

7.34 

3.19 

7.32 

3.22 

8 

9 

8.28 

3.52 

8.27 

3.55 

8.25 

3.59 

8.24 

3.62 

9 

10 

9.20 

3.91 

9.19 

3.95 

9.17 

3.99 

9.15 

4.03 

10 

11 

10.13 

4.30 

10.11 

4.34 

10.09 

4.39 

10.07 

4.43 

11 

12 

11.05 

4.69 

11.03 

4.74 

11.00 

4.78 

10.98 

4.83 

12 

13 

11.97 

6.08 

11.94 

5.13 

11.92 

5.18 

11.90 

5.24 

13 

14 

12.89 

6.47 

12.86 

5.53 

12.84 

5.58 

12.81 

5.64 

14 

15 

13.81 

5.86 

13.78 

5.92 

13.76 

5.98 

13.73 

6.04 

15 

16 

14.73 

6.25 

14.70 

6.32 

14.67 

6,38 

14.64 

6.44 

16 

17 

15.65 

6.64 

15.62 

6.71 

15.59 

6.78 

15.56 

6.85 

17 

18 

16.,  57 

7.03 

16.54 

7.11 

16.51 

7.18 

16.48 

7.25 

18 

19 

17.49 

7.42 

17.46 

7.50 

17.42 

7.58 

17.39 

7.65 

19 

20 

18.41 

7.81 

18.38 

7.89 

18.34 

7.97 

18.31 

8.05 

20 

21 

19.33 

8.21 

19.29 

8.29 

19.26 

8.37 

19.22 

8.46 

21 

22 

20.25 

8.60 

20.21 

8.68 

20.18 

8.77 

20.14 

8.86 

22 

23 

21.17 

8.99 

21.13 

9.  -08 

21.09 

9.17 

21.05 

9.26 

23 

24 

22.09 

9.28 

22.05 

9.47 

22.01 

9.57 

21.97 

9.67 

24 

25 

23.01 

9.77 

22.97 

9.87 

22.93 

9.97 

22.88 

10.07 

25 

26 

23.93 

10.16 

23.89 

10.26 

23.84 

10.37 

23.80 

10.47 

26 

27 

24.85 

10.55 

24.81 

10.66 

24.76 

10.77 

24.71 

10.87 

27 

28 

25.77 

10.94 

25.73 

11.05 

25.68 

11.16 

25.63 

1)  .28 

28 

29 

26.69 

11.33 

26.64 

11.45 

26.59 

11.56 

26.54 

11.68 

29 

30 

27.62 

11.72 

27.56 

11.84 

27.51 

11.96 

27.46 

12.08 

30 

31 

28.54 

12.11 

28.48 

12.24 

28.43 

12.36 

28.37 

12.49 

31 

32 

29.46 

12.50 

29.40 

12.63 

29.35 

12.76 

29.29 

12.89 

3,2 

33 

30.38 

12.89 

30.32 

13.03 

30.26 

13.16 

30.21 

13.29 

33 

34 

31.30 

13.28 

31.24 

13.42 

31.18 

13.56 

31.12 

13.69 

34 

35    32.22 

13.68 

32.16 

13.82 

32.10 

13.96 

32.04 

14.10 

35 

36 

33,  14 

14.07 

33.08 

14.21 

33.01 

14.35 

32.95 

14.50 

36 

37 

34.06 

14.46 

34.00 

14.61 

33  .  93 

14.75 

33.87 

14.90 

37 

38 

34.98 

14.85 

34.91 

15.00 

34.85 

15.15 

34.78 

15.30 

38 

39 

35.90 

15.24 

35.83 

15.39 

35.77 

15.55 

35.70 

15.71 

39 

40 

36.82 

15.63 

36.75 

15.79 

36.68 

15.95 

36.61 

16.11 

40 

41 

37.74 

16.02 

37.67 

16.18 

37  .  60 

16.35 

37.53 

16.51 

41 

42 

38.66 

16.41 

38.59 

16.58 

38.52 

16.75 

38.44 

16.92 

42 

43 

39.58 

16.80 

39.51 

16.97 

39.43 

17.15 

39.36 

17.32 

43 

44 

40.50 

17.19 

40.43 

17.37 

40.35 

17.54 

40.27 

17.72 

44 

45 

41.42 

17.58 

41.35 

17.76 

41.27 

17.94 

41.19 

18.12 

45 

46 

42.34 

17.97 

42.26 

18.16 

42.18 

18.34 

42.10 

18.53 

46 

47 

43.26 

18.36 

43.18 

18.56 

43.10 

18.74 

43.02 

18.93 

47 

48 

44.18 

18.76 

44.10 

18.95 

44.02 

19.14 

43  .  93 

19.33 

48 

49 

45.10 

19.15 

45.02 

19.34 

44.94 

19.54 

44.85 

19.73 

49 

50    46.03    19.54 

45.94 

19.74 

45.85 

19.94 

45.77 

20.14 

50 

§      Dep. 

2  i 

Lat. 

Dep.      Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

B 

'   1 

I 
Q 

67  Deg. 

66|  Deg. 

66^  Deg. 

66$  Deg. 

Q 

TRAVERSE    TABLE. 


49 


- 

23  Deg. 

234  Deg. 

23£  Deg. 

23|  Deg. 

D 

3 
O 
O 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  |  Dep. 

3 
o 

51 

46.95 

19.93 

46.86 

20.  13 

46.77 

20.34 

46.68 

20.54 

51 

52 

47.87 

20  .  32 

47.78 

20  .  53 

47.69 

20.73 

47.60 

20.94 

52 

53 

43.79 

20.71 

48.70 

20.92 

48.60 

•21.13 

48.51 

21.35 

53 

54 

49.71 

21.10 

49.61 

21.32 

49.52 

21.53 

49.43 

21.75 

54 

55 

50  .  63 

21.49 

50.53 

•21.71 

50.44 

21.93 

50.34 

22.15 

55 

56 

51.55 

21.88 

51.45 

22  .  1  1 

51.36 

22.33 

51.26 

22.55 

56 

57 

52.47 

22.27 

52.37 

22.50 

52.27 

22.73 

52.17 

22.96 

57 

53 

53.39 

22.66 

53.29 

22.90 

53.19 

23.13 

53.09 

23.36 

58 

59 

54.31 

23.05 

54.21 

23.29 

54.11 

23.53 

54.00 

23.76 

59 

60 

55.23 

23.44 

55.13 

23.68 

55.02 

23.92 

54.92 

24.16 

60 

61 

56  .  1-5 

23.83 

56.05 

24.08 

55.94 

24.32 

55.83 

24.57 

61 

62 

57.07 

24.23 

56.97 

24.47' 

56.86 

24.72 

56.75 

24.97 

62 

63 

57.99 

24.62 

57.88 

24.87 

57.77 

25.12 

57.66 

25.37 

63 

64 

5.3.91 

25.01 

58.80 

25.26 

58.69 

25.52 

58.58 

25.78 

64 

65 

59.83 

25.40 

59.72 

25.66 

59.61 

25.92 

59.50 

26.18 

65 

66 

60.75 

2-5.79 

60.64 

26.05 

60.53 

26  .  32 

60.41 

26.58 

66 

•67 

61.67 

26.18 

61.56 

25.45 

61.44 

26.72 

61.33 

26.98 

67 

68 

62.59 

26.57 

62.48 

26.84 

62.36 

27.11 

62.24 

27.39 

68 

69 

63.51 

26.96 

63.40 

27.24 

63.28 

27.51 

63.16 

27.79 

69 

70 

64.44 

27.35 

64.32 

27.63 

64.19 

27.91 

64.07 

28.19 

70 

71- 

65.36 

27.74 

65.23 

28.03 

65.11 

28.31 

64.99 

23.59 

71 

72 

66.28 

28.13 

66.15 

28.42 

66.03 

28.71 

65.90 

29.00 

72 

73 

67.20 

28.52 

67.07 

28.82 

66.95 

29.11 

66.82 

29.40 

73 

74 

68.12 

28.91 

67.99 

29.21 

67.86 

29.51 

67.73 

29.80 

74 

75 

69.04 

29.30 

68.91 

29.61 

68.78 

29.91 

68.65 

30.21 

75 

76 

69.96 

29.70 

69.83 

30.00 

69.70 

30.30 

69.56 

30.61 

76 

77 

70.88 

30.09 

70.75 

30.40 

70.61 

30.70 

70.48 

31.01 

77 

73 

71.30 

30.48 

71.67 

30.79 

71.53 

31.10 

71.39 

31.41 

73 

79 

72.72 

30.87 

72.58 

31.18 

72.45 

31.50 

72.31 

31.82 

79 

80 

73.64 

31.26 

73.50 

31.58 

73.36 

31.90 

73.22 

32.22 

80 

81 

74.56 

31.65 

74.42 

31.97 

74.23 

32.30 

74.14 

32.62 

81 

8-4    75.48 

32.04 

75.34 

32.37 

75.20 

32.70 

75.06 

33.03 

82 

83    76.40 

32.43 

76.26 

32.76 

76.12 

33.10 

75.97 

33.43 

83 

S4    77.32 

32.82 

77.18 

33.16 

77.03 

33.49 

76.89 

33.83 

84 

85    78.24 

33.21 

78.10 

33  .  55 

77.95 

33.89 

77.80 

34.23 

85 

fifi    79.16 

33.60 

79.02 

33.95 

78.87 

34.29 

78.72 

34.64 

86 

87    80.08 

33.99 

79.93 

34.34 

79.78 

34.69 

79.63 

35.04 

87 

83    81.00 

34-.  38 

80.85 

34.74 

80.70 

35.09 

80.55 

35.44 

88 

89    81.92 

34.78 

81.77 

35.13 

81.62 

35.49 

81.46 

35.84 

89 

«JO    82.85 

35.17 

32.69 

35.53 

82.54    35.89 

82.38 

36.25 

90 

91    83.77 

35.56 

83.61 

35.92! 

83.451  36.29 

83.29 

36.65 

91 

92    84.69 

35.95 

84.53 

36.32 

84.37 

36.68 

84.21 

37.05 

92 

93    85.61 

36.34 

85.45 

36.71 

85.29 

37.08 

85.12 

37.46 

93 

94  186.53 

36.73 

86.37 

37.11 

86.20 

37.48 

86.04 

37.86 

94 

95  i  87.  45    37.12 

37.29 

37.501 

87.12 

37.88| 

86.95 

38.26 

95 

96    88.37    37.51 

88.20 

37.901 

88.04    38.28  j 

87.87 

38.66 

96 

97    89.29    37.90 

89.12 

38.29 

88.95 

38.68 

88.79 

39.07 

97 

98190.21    38.29 

90.04 

33.68 

89.87 

39.08 

89.70 

39.47 

98 

99    91.13    38.68 

90.96 

39.08! 

90.79 

39.48 

90.62 

39.87 

99 

100    92.05    39.07 

91.88 

39.47i 

91.71 

39.87 

91.50 

40.27 

100 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o> 
q 
a 

3 

67  Deg. 

66|  Deg. 

661  Deg. 

66*  Deg. 

q 

TRAVERSE    TABLE. 


B' 

24  Deg. 

244  Deg. 

24^  Deg. 

24|  Deg. 

G 

05* 

1 

I 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

E 
8 

1 

0.91 

0.41 

0.91 

0.41 

0.91 

0.41 

0.91 

0.42 

1 

2 

1.83 

0.81 

1.82 

0.82 

1.82 

0.83 

1.82 

0.84 

2 

3 

2,74 

1.22 

2.74 

1.23 

2.73 

1.24 

2.72 

1.26 

4 

3.65 

1.63 

3.65 

1.64 

3.64 

1.66 

3.63 

1.67 

4 

5 

4.57 

2.03 

4.56 

2.05 

4.55 

2.07 

4.54 

2.09 

5 

J6 

5.48 

2.44 

5.47 

2.46 

5.46 

2.49 

5.45 

2.51 

6 

7 

6.39 

2.85 

6.38 

2.87 

6.37 

.  2.90 

6.36 

2.93 

7 

8 

7.31 

3.25 

7.29 

3.29 

7.28 

3.32 

7.27 

3.35 

8 

9 

8.22 

3.66 

8.21 

3.70 

8.19 

3.73 

8.17 

G.77 

9 

10 

9.14 

4.07 

9.12 

4.11 

9.10 

4.15 

9.08 

4.19 

10 

11 

10.05 

4.47 

10.03 

4.52 

10.01 

4.56 

9.99 

4.61 

11 

12 

10.96 

4.88 

10.94 

4.93 

10.92 

4.98 

10.90 

5.02 

12 

13 

11.88 

5.29 

11.85 

5.34 

11.83 

5.39 

11.81 

5.44 

13 

14 

12.79 

5.69 

12.76 

5.75 

12.74 

5.81 

12.71 

5.86 

14 

15 

13.70 

6.10 

13.68 

6.16 

13.65 

6.22 

13.62 

6.28 

15 

16 

14*.  62 

6.51 

14.59 

6.57 

14.56 

6.64 

14.53 

6.70 

16 

17 

15.53 

6.  92 

15.50 

6.98 

15.47 

7.05 

15.44 

7.12 

17 

18 

16.44 

7.32 

16.41 

7.39 

16.38 

7.46 

K6.35 

7.54 

18 

19 

17.36 

7.73 

17.32 

7.80 

17.29 

7.88 

17.25 

7.95 

19 

20 

18.27 

8.13 

18.24 

8.21 

18.20 

8.29 

18.16 

8.37 

20 

2i 

19.18 

8.54 

19.15 

8.63 

19.11 

8.71 

19.07 

8.79 

21 

22 

20.10 

8.95 

20.06 

9.04 

20.02 

9.12 

19.98 

9.21 

22 

23 

21.01 

9.35 

20.97 

9.45 

20.93 

9.54 

20.89 

9.63 

23 

24 

21.93 

9.76 

21.88 

9.86 

21.84 

9.95 

21.80 

10.05 

24 

25 

22.84 

10.17 

22.79 

10.27 

22.75 

10.37 

22.70 

10.47 

25 

26 

23.75 

10.58 

23.71 

10.68 

23.66 

10.78 

23.61 

10.89 

26 

27 

24.67 

10.98 

24.62 

11.09 

24.57 

11.20 

24.52 

11.30 

27 

28 

25.58 

11.39 

25.53 

11.50 

25.48 

11.61 

25.43 

11.72 

28 

29 

26.49 

11.80 

26.44 

11.91 

26.39 

12.03 

26.34 

12.14 

29 

30 

27.41 

12.20 

27.35 

12.32 

27.30 

12.44 

27.24 

12.56 

30 

31 

28.32 

12.61 

28.26 

12.73 

28.21 

12.86 

28.15 

12.98 

31 

32 

29.23 

13.02 

29.18 

13.14 

29.12 

13.27 

29.06 

13.40 

32 

33 

30.15 

13.42 

30.09 

13.55 

30.03 

13.68 

29.97 

13.82 

33 

34 

31.06 

13.83 

31.00 

13.96 

30.94 

14.10 

30.88 

14.23 

34 

35 

31.97 

14.24 

31.91 

14.38 

31.85 

14.51 

31.78 

14.65 

35 

36 

32.89 

14.64 

32.82 

14.79 

32.76 

14.93 

32.69 

15.07 

36 

37 

33.80 

15.05 

33.74 

15.20 

33.67 

15.34 

33.60 

15.49 

37 

38 

34.71 

15.46 

34.65 

15.61 

34.58 

15.76 

34.51 

15.91 

38 

39 

35.63 

15.86 

35.56 

16.02 

35.49 

16.17 

35.42 

16.33 

39 

40 

36.54 

16.27 

36.47' 

16.43 

36.40 

16.59 

36.33 

16.75 

40 

41 

37.46 

16.68 

37.38 

16.84 

37.31 

17.00 

37.23 

17cl6 

*  41 

42 

38.37 

17.08 

38  .  29 

17.25 

38.22 

17.42 

38.14 

17.58 

42 

43 

39.28 

17.49 

39.21 

17.66 

39.13 

17.83 

39.05 

18.00 

43 

44 

40.20 

17.90 

40.12 

18.07 

40.04 

18.25 

39.96 

18.42 

44 

45 

41.11 

18.30 

41.03 

18.48 

40.95 

18.66 

40.87 

18.84 

45 

46 

42.02 

18.71 

41.94 

18.89 

41.86 

19.08 

41.77 

19.26 

46 

47 

42  .  94 

19.12 

42.85 

19.30 

42.77 

19.49 

42.68 

19.68 

47 

48 

43.85 

19.52 

43.76 

19.71 

43.68 

19.91 

43.59 

20.10 

48 

49 

44.76 

19.93 

44.68 

20.13 

44.59 

20.32 

44.50 

20.51 

49 

50 

45  .  68 

20.34 

45.59 

20.54 

45.50 

20.73 

45.41 

20.93 

50 

8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

10 

o 

"oi 

d 

d 
to 

5 

66  Deg. 

65|  Deg. 

65|  Deg. 

654  I>eg. 

s 

TRAVERSE    TABLE. 


51 


o 

24  Deg. 

24i  Deg. 

24*  Deg. 

24|  Deg. 

O 

P 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

? 

51 

46.59 

20.74 

46.50 

20.95 

46.41 

21.15 

46.32 

21.35 

51 

52 

47.50 

21.15 

47.41 

21.36 

47.32 

21.56 

47.22 

21.77 

52 

53 

48.42 

21.56 

48.32 

21.77 

48.23 

21.98 

4*.  13 

22.19 

53 

54 

49.33 

21.96 

49.24 

22.18 

49.14 

22.39 

49.04 

22.61 

54 

55 

50.24 

22.37 

50.15 

22.59 

50.05 

22.81 

49.95 

23.03 

55 

56 

51.16 

22.78 

51.06 

23.00 

50.96 

23.22 

50.86 

23.44 

56 

57 

52.07 

23.18 

51.97 

23.41 

51.87 

23.64 

51.76 

23.86 

57 

58 

52.99 

23.59 

52.88 

23.82 

52.78 

24.05 

52.67 

24.28 

58 

59 

53.90 

24.00 

53.79 

24.23 

53.69 

24.47 

53.58 

24.70 

59 

60 

54.81 

24.40 

54.71 

24  .  64 

54.60 

24.88 

54.49 

25.12 

60 

61 

55.73 

24.81 

55.62 

25.05 

55.51 

25.30 

55.40 

25.54 

61 

62 

56.64 

25.22 

56.53 

25.46 

56.42 

25.71 

56.30 

25.96 

62 

63 

57.55 

25.62 

57.44 

25.88 

57.33 

26.13 

57.21 

26.38 

63 

64 

58.47 

26.03 

58.35 

26.29 

58.24 

26.54 

58.12 

26.79 

64 

65 

59.38 

26.44 

59.26 

26.70 

59.15 

26.96 

59.03 

27.21 

65 

66 

60.29 

26.84 

60.18 

27.11 

60.06 

27.37 

59.94 

27.63 

66 

67 

61.21 

27.25 

61.09 

27.52 

60.97 

27.78 

60.85 

28.05 

67 

68 

62.12 

27.66 

62.00 

27.93 

61.88 

28.20 

61.75 

28.47 

68 

69 

63.03 

28.06 

62.91 

28.34 

62.79 

28.61 

62.66 

28.89 

69 

70 

63.95 

28.47 

63.82 

28.75 

63.70 

29.03 

63.57 

29.31 

70 

71 

64.86 

28.88 

64.74 

29.16 

64.61 

29.44 

64.48 

29.72 

71 

72 

65.78 

29.28 

65.65 

29.57 

65.52 

29.86 

65.39 

30.14 

72 

73 

66.69 

29.69 

66.56 

29.98 

66.43 

30.27 

66.29 

30.56 

73 

74 

67.60 

30.10 

67.47 

30.39 

67.34 

30.69 

67.20 

30.98 

74 

75 

68.52 

30.51 

68.38 

30.80 

68.25 

31.10 

68.11 

31.40 

75 

76 

69.43 

30.91 

69.29 

31.21 

69.16 

31.52 

69.02 

31.82 

76 

77 

70.34 

31.32 

70.21 

31.63 

70.07 

31.93 

69.93 

32.24 

77 

78 

71.26 

31.73 

71.12 

32.04 

70.98 

32.35 

70.84 

32.66 

78 

79 

72.17 

32.13 

72.03 

32.45 

71.89 

32.76 

71.74 

33.07 

79 

80 

73.08 

32.54 

72.94 

32.86 

72.80 

33.18 

72.65 

33.49 

80 

81 

74.00 

32.95 

73.85 

33.27 

73.71 

33.59 

73.56 

33.91 

81 

82 

74.91 

33.35 

74.76 

33.68 

74.62 

34.00 

74.47 

34.33 

82 

83 

75.82 

33.76 

75.68 

34.09 

75.53 

34.42 

75.38 

34.75 

83 

84 

76.74 

34.17 

76.59 

34.50 

76.44 

34.83 

76.28 

35.17 

84 

85 

77.65 

34.57 

77.50 

34.91 

77.35 

35.25 

77.19 

35.59 

85 

86 

78.56 

34.98 

78.41 

35.32 

78  26 

35.66 

78.10 

36.00 

86 

87 

79.48 

35.39 

79.32 

35.73 

79.17 

36.08 

79.01 

36.42 

87 

88 

80.39 

35.79 

80.24 

36.14 

80.08 

36.49 

79.92 

36.84 

88 

89 

81.31 

36.20 

81.15 

36.55 

80.99 

36.91 

80.82 

37.26 

89 

90 

82.22 

36.61 

82.06 

36.96 

81.90 

37.32 

81.73 

37.68 

90 

91 

83.13 

37.01! 

82.97 

37.38 

82.81 

37.74 

82.64 

33.10 

91 

92 

84.05 

37.42 

83.88 

37.79 

83.72 

38.15 

83.55 

38.52 

92 

93 

84.96 

37.83 

84.79 

38.20 

84.63 

38.57 

84.46 

38.94 

93 

94 

85.87 

38.23 

85.71 

38.61 

85.54 

38.98 

85.37 

39.35 

94 

95 

86.79 

38.64 

86.62 

39.02 

86.4,5 

39.40 

86.27 

39.77 

95 

96 

87.70 

39.051 

87.53 

39.43 

87.36 

39.81 

87.18 

40.19 

96 

97 

88.61 

39.45 

88.44 

39.84 

88.27 

40.23 

88.09 

40.61 

97 

98 

89.53 

39.86 

89.35 

40.25 

89.18 

40.64 

89.00 

41.03 

98 

99  !  90.44 

40.27 

90.26 

40.66 

90.09 

41.05 

89.91 

41.45 

99 

100 

91.35 

40.67i 

91.18 

41.07 

91.00 

41.47 

90.81 

41.87 

100 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

~i 

Q 

66  Deg. 

65|  Deg. 

65i  Deg. 

654  Deg. 

rt 
7c 

Q 

TRAVERSE    TABLE. 


c 

25  Deg. 

25}  Deg. 

25^  Deg. 

25|  Deg. 

| 

1 

£ 

B 

o 

Lat. 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

~1 

0.91 

0.42 

O.CO 

0.43 

0.90 

0.43 

0.90 

0.43 

i 

2 

1.81 

0.85 

1.81 

0.85 

1.81 

0.86 

1.80 

0.87 

2 

3 

2.72 

1.27 

2.71 

1.28 

2.71 

1.29 

2.70 

1.30 

3 

4 

3.63 

1.69 

3.62 

1.71 

3.61 

1.72 

3.60 

1.74 

4 

5 

4.53 

2.11 

4.52 

2.13 

4.51 

2.15 

4.50 

2.17 

5 

6 

5.44 

2.54 

5.43 

2.56 

5.42 

2.58 

5.40 

2.61 

6 

7 

6.34 

2.96 

6.33 

2.99 

6.32 

3.01 

6.30 

3.04 

7 

8 

7.25 

3.38 

7.24 

3.41 

7.22 

3.44 

7.21 

3.48 

8 

9 

8.16 

3.80 

.8.14 

3.84 

8.12 

3.87 

8.11 

3.91 

9 

10 

9.06 

4.23 

9.04 

4.27 

9.03 

4.31 

9.01 

4.34 

10 

11 

9.97 

4.65 

9.95 

4.69 

9.93 

4.74 

9.91 

4.78 

11 

12 

10.88 

5.07 

10.85 

5.12 

10.83 

5.17 

10.81 

5.21 

12 

13 

11.78 

5.49 

11.76 

5.55 

11.73 

5.60 

11.71 

5.65 

13 

14 

12.69 

5.92 

12.66 

5.97 

1^.64 

6.03 

12.61 

6.08 

14 

15 

13.59 

6.34 

13.57 

6.40 

13.54 

6.46 

13.51 

6.52 

15 

16 

14.50 

6.76 

14.47 

6.83 

14.44 

6.89 

14.41 

6.95 

16 

17 

15.41 

7.18 

15.38 

7.25 

15.34 

7.32 

15.31 

7.39 

17 

18 

16.31 

7.61 

16.28 

7.68 

16.25 

7.75 

16.21 

7.82 

18 

19 

17.22 

8.03 

17.18 

8.10 

17.15 

8.18 

17.11 

8.25 

19 

20 

18.13 

8.45 

18.09 

8.53 

18.05 

8.61 

18.01 

8.69 

20 

21 

19.03 

8.87 

18.99 

8.96 

18.95 

9.04 

18.91 

9.12 

21 

22 

19.94 

9.30 

19.90 

9.38 

19.86 

9.47 

19.82 

9.56 

22 

23 

20.85 

9.72 

20.80 

9.81 

20.76 

9.90 

20.72 

9.99 

23 

24 

2J.75 

10.14 

21.71 

10.24 

21.66 

10.33 

21.62 

10.43 

24 

25 

22.66 

10.57 

22.61 

10.66 

22.56 

10.76 

22.52 

10.86 

25 

26 

23.56 

10.99 

23.52 

11.09 

23.47 

11.19 

23.42 

11.30 

26 

27 

24.47 

11.41 

24.42 

11.52 

24.37 

11.62 

24.32 

11.73 

27 

28 

25.38 

11.83 

25.32 

11.94 

25.27 

12.05 

25.22 

12.16 

28 

29 

26.28 

12.26 

26.23 

12.37 

26.17 

12.48 

26.12 

12.60 

29 

30 

27.19 

12.68 

27.13 

12.80 

27.08 

12.92 

27.03 

13.03 

30 

31 

28.10 

13.10 

28.04 

13.22 

27.98 

13.35 

27.92 

13.47 

31 

32 

29.00 

13.52' 

28.94 

13.65 

28.88 

13.78 

28.82 

13.90 

32 

33 

29.91 

13.95 

29.85 

14.08 

29.79 

14.21 

29.72 

14.34 

33 

34 

30.81 

14.37 

30  .75 

14.50 

30.69 

14.64 

30.62 

14.77 

34 

35 

31.72 

14.79 

31.66 

14.93 

31.59 

15.07 

31.52 

15.21 

35 

36 

32.63 

15.21 

32.56 

15.36 

32.49 

15.50 

32.43 

15.64 

37 

33.53 

15.64 

33.46 

15.78 

33.40 

15.93 

33.33 

16.07 

f 

38 

34.44 

16.06 

34.37 

16.21 

34.30 

16.36 

34.23 

16.51 

39 

35.35 

16.48 

35.27 

16.64 

35.20 

16.79 

35.13 

16.94 

* 

40 

36.25 

16.90 

36.18 

17.06 

36.10 

17.22 

36.03 

17.38 

4 

41 

37.16 

17.33 

37.08 

17.49 

37.01 

17.65 

36.93 

17.81 

42 

38.06 

17.75 

37.99 

17.92 

37.91 

18.08 

37.83 

18.25 

/ 

43 

38.97 

18.17 

38.89 

18.34 

38.81 

18.51 

38.73 

18.68 

44 

39.88 

18.60 

39.80 

18.77 

39.71 

18.94 

39.63 

19.12 

45 

40.78 

19.02 

40.70 

19.20 

40.62 

19.37 

40.53 

19.55 

46 

41.69 

19.44 

41.60 

19.62' 

41.52 

19.80 

41.43 

19.98 

^ 

47 

42.60 

19.86 

42.51 

20.05 

42.42 

20.23 

42.83 

20.42 

48 

43.50 

20.29 

43.41 

20.48 

43.32 

20.66 

43,23 

20.85 

L 

49 

44.41 

20.71 

44.32 

20.90 

44.23 

21.10 

44.13 

21.29 

i 

50 

45.32 

21.13 

45.22 

21.33 

45.13 

21.53 

45.03 

21.72 

5 

I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Is 

1 

65  Deg. 

64J  Deg. 

64i  Deg. 

644  Deg. 

TKAVERSE    TABLE. 


53 


c 

£' 

25  Deg. 

25i  Deg. 

254  Deg. 

25  1  Deg. 

D 

lance.  1 

Lat. 

stance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Dep. 

51 

46.22 

21.55 

46.13 

21.75 

46.03 

21.96 

45.94 

22.16 

51 

52 

47.13 

21.98 

47.03 

22.18 

46.93 

22.39 

46.84 

22.*>9 

52 

53 

48.03 

22.40 

47.94 

22.61 

47.84 

22.82 

47.74 

23.03 

53 

54 

48.94 

22.82 

48.84 

23.03 

48.74 

23.25 

48.64 

23.46 

54 

55 

49.85 

23.24 

49.74 

23.46 

49.64 

23.68 

49.54 

23.89 

55 

56 

50.75 

23.67 

50.65 

23.89 

50.54 

24.11 

50.44 

24.33 

56 

57 

51.66 

24.09 

51.55 

24.31 

51.45 

24.54 

51.34 

24.76 

57 

58 

52.57 

24.51 

52.46 

24.74 

52.35 

24.97 

52.24 

25.20 

58 

59 

53.47 

24.93 

53,36 

25.17 

53.25 

25.40 

53.14 

25  .  63 

59 

60 

54.38 

25.36 

54.27 

25.59 

54.16 

25.83 

54.04 

26.07 

60 

61 

55.28 

25.78 

55.17 

26.02 

55.06 

26.26 

54.9-1 

26.50 

61 

62 

56.19 

26.20 

56.08 

26.45 

55.96 

26.69 

55.84 

26.94 

62 

63 

57.10 

26.62 

56  .  98 

26.87 

56.86 

27.12 

56  .  74 

27.37 

63 

64 

58.00 

27.05 

57.89 

27.30 

57.77 

27.55 

57.64 

27.80 

64 

65 

58.91 

27.47 

58.79 

27.73 

58.67 

27.98 

58.55 

28.24 

65 

66 

59.82 

27.89 

59  .  69 

28.15 

59  .  57 

28.41 

59.40 

28.67 

66 

67 

60.72 

28.32 

60.60 

28.58 

60.47 

28.84 

60.35 

29.11 

67 

68 

61.63 

28.74 

61.50 

29.01 

61.38 

29.27 

61.25 

29.54 

68 

69 

62.54 

29.16 

62.41 

29.43 

62.28 

29.71 

62.15 

29.98 

69 

70 

63.44 

29.58 

63.31 

29.86 

63.18 

30.14 

63.05 

30.41 

70 

71 

64.35 

30.01  i 

64.22 

30.29 

64.08 

30.57 

63.95 

30.85 

71 

72 

65.25 

30.43 

65.12 

30.71 

64.99 

31.00 

64.85 

31.28 

72 

73 

66.16 

30.85  j 

66.03 

31.14 

65.89 

31.43 

65.75 

31.71 

73 

74 

67.07 

31.27 

66.93 

31.57 

66.79 

31.86 

66.65 

32.15 

74 

75 

67.97 

31.70 

67.83 

31.99 

67.69 

32.29 

67.55 

32.58 

75 

76 

68.88 

32.12 

68.74 

32.42 

68.60 

32.72 

68.45 

33.02 

76 

77 

69.79 

32.54 

69.64 

32.85 

69.50 

33.15 

69.35 

33.45 

77 

78 

70.69 

32.96 

70.55 

33.27 

70.40 

33.58 

70.25 

33.89 

78 

79 

71.60 

33.39 

71.45 

33.70 

71.30 

34.01 

71.16 

34.32 

79 

80 

72.50 

33.81 

72.36 

34,13 

72.21 

34.44 

72  06 

34.76 

80 

81 

73.41 

34.23 

73.26 

34.55 

73.11 

34.87 

72.96 

35.19 

81 

82 

74.32 

34.65 

74.17 

34.98 

74.01 

35  .  30 

73.86 

35.62 

82 

83 

75.22 

35.08 

75.07 

35.41 

74.91 

35.73 

74.76 

38.06 

83 

84 

76.13 

35  .  50 

75.97 

35.83 

75.82 

36.16 

75.66 

36.49 

84 

85 

77.04 

35.92 

76.88 

36.26 

76.72 

36.59 

76.56 

36.93 

85 

86 

77.94 

36.35 

77.78 

36.68 

77.62 

37.02 

77.46 

37.36 

86 

87 

78.85 

36.77 

78.69 

37.11 

78.52 

37.45 

78.36 

37.80 

87 

88 

79.76 

37.19 

79.59 

37.54 

79.43 

37.88 

79.26 

38.23 

88 

89 

80.66 

37.61 

80;  50 

37.96 

80.33 

38.32 

80.16 

38.67 

89 

90 

81.57 

38.04 

81.40 

38.39 

81.23 

38.75 

81.06 

39.10 

90 

91 

82.47 

38.46 

82.31 

38.82  82.14 

39.18 

81.96 

39.53 

91 

92  83.38 

38.88 

83.21 

39.24  83.04 

39.61 

82.86 

39.97 

92 

93 

84.29 

39.30 

84.11 

39.67 

83.94 

40.04 

83.76 

40.40 

93 

94 

•So.  19 

39  .  73 

85.02 

40.10 

84.84- 

40.47 

84.67 

40.84 

94 

95 

86.10 

40.15 

85.92 

40.52 

85.75 

40.90 

85.57 

41.27 

95 

96 

87.01 

40.57 

86.83 

40.95  186.65 

41.33 

86.47 

41.71 

96 

97 

87.91 

40.99 

87.73 

41.38  87.55 

41.76 

87.37 

42.14 

97 

98 

88.82 

41.42 

88.64 

41.80  88.45 

42.19 

88.27 

42.58 

98 

99 

89.72 

41.84 

89.54 

42.23  89.36 

42.62 

89.17 

43.01 

99 

100 

90.63 

42.26 

90.45 

42.66  ;90.26 

43.05 

90.07 

43.44 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

<»' 
o 

c 

I 

a 

65  Deg. 

641  Deg. 

64i  Deg. 

64J  Deg. 

J2 

.2 
3 

i 

54 


TRAVERSE    TABLE. 


o 

5° 

26  Deg. 

264  Deg. 

26  £  Deg. 

26|  Deg. 

O 
K- 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

0.90 

0.44 

0.90 

0.44 

0.89 

0.45 

0.89 

0.45 

1 

2 

1.80 

0.88 

1.79 

0.88 

1.79 

0.89 

1.79 

0.90 

2 

3 

2.70 

1.32 

2.69 

1.33 

2.68 

1.34 

2.68 

1.35 

3 

4 

3.60 

1.75 

3.59 

1.77 

3.58 

1.78 

3.57 

1.80 

4 

5 

4.49 

2.19 

4.48 

2.21 

4.47 

2.23 

4.46 

2.25 

5 

6 

5.39 

2.63 

5.38 

2.65 

5.37 

2.68 

5.36 

2.70 

6 

7      6.29 

3.07 

6.28 

3.10 

6.26 

3.12 

6.25 

3.15 

7 

8 

7.19 

3.51 

7.17 

3.54 

7.16 

3.57 

7.14 

3.60 

8 

9 

8.09 

3.95 

8.07. 

3.98 

8.05 

4.02 

8.04 

4.05 

9 

10 

8.99 

4.38 

8.97 

4.42 

8.95 

4.46 

8.93 

4.50 

10 

11 

9.89 

4.82 

9.87 

4.87 

9.84 

4.91 

9.82 

4.95 

11 

12 

10.79 

5.26 

10.76 

5.31 

10.74 

5.35 

10.72 

5.40 

12 

13 

11.68 

5.70 

11.66 

5.75 

11.63 

5.80 

11.61 

5.85 

13 

14 

12.58 

6.14 

12.58 

6.19 

12.53 

6.25 

12.50 

6.30 

14 

15 

13.48 

6.58 

13.45 

6.63 

13.42 

6.69 

13.39 

6.75 

15 

16 

14.38 

7.01 

14.35 

7.08 

14.32 

7.14 

14.29 

7.20 

16 

17 

15.28 

7.45 

15.25 

7.52 

15.21 

7.59 

15.18 

7.65 

17 

18 

16.18 

7.89 

16.14 

7.96 

16.11 

8.03 

16.07 

8.10 

18 

19 

17.08 

8.33 

17.04 

8.40 

17.00 

8.48 

16.97 

8.55 

19 

20 

17.98 

8.77 

17.94 

8.85 

17.90 

8.92 

17.86 

9.  GO 

20 

21 

18.87 

9.21 

18.83 

9.29 

18.79 

9.37 

18.75 

9.45 

21 

22 

19.77 

9.64 

19.73 

9.73 

19.69 

9.82 

19.65 

9.90 

22 

23 

20.67 

10.08 

20.63 

10.17 

20.58 

10.26 

20.54 

10.35 

23 

24 

21.57 

10.52 

21.52 

10.61 

21.48 

10.71 

21.43 

10.80 

24 

25 

22.47 

10.96 

22.42 

11.06 

22.37 

11.15 

22.32 

11.25 

25 

26 

23.37 

11.40 

23.32 

11,50 

23.27 

1  1  .  60 

23.22 

11.70 

26 

27 

24.27 

11.84 

24.22 

11.94 

24.16 

12.05 

24.11 

12.15 

27 

28 

25.17 

12.27 

25.11 

12.38 

25.06 

12.49 

25.00 

12.60 

28 

29  {26.06 

12.71 

26.01 

12.83 

25  .  95 

12.94 

25.90 

13.05 

29 

30  126.96 

13.15 

26.91 

13.27 

26.85 

13.39 

26.79 

13.50 

30 

31 

27.86 

13.59 

27.80 

13.71 

27.74 

13.83 

27.68 

13.95 

31 

32 

28.76 

14.03 

28.70 

14.15 

28.64 

14.28 

28.58 

14.40 

32 

33 

29.66 

14.47 

29.60 

14.60 

29.53 

14.72 

29.47 

14.85 

33 

34 

30.56 

14.90 

30.49 

15.04 

30.43 

15.17 

30.36 

15.30 

34 

35 

31.46 

15.34 

31.39 

15.48 

31.32 

15.62 

31.25 

15.75 

35 

36 

32.36 

15.78 

32.29 

15.92 

32.22 

16.06 

32.15 

16.20 

36 

37 

33.26 

16.22 

33.18 

16.36 

33.11 

16.51 

33.04 

16.65 

37 

38 

34.15 

16.66 

34.08 

16.81 

34.01 

16.96 

33.93 

17.10 

38 

39 

35.05 

17.10 

34.98 

17.25 

34.90 

17.40 

34.83 

17.55 

39 

40 

35.95 

17.53 

35  .  87 

17.69 

35.80 

17.85 

35.72 

18.00 

40 

41 

36.85    17.97 

36.77 

18.13 

36.69 

18.29 

36.61 

18.45 

41 

42 

37.75 

18.41 

37.67 

18.58 

37.59 

18.74 

37.51 

18.90 

42 

43 

38.65 

18.85 

38.57 

19.02 

38.48 

19.19 

38.40 

19.35 

43 

44 

39.55 

19.29 

39.46 

.19.46 

39.38 

19.63 

39.29 

19.80 

44 

45 

40.45 

19.73 

40.36 

19.90 

40.27 

20.08 

40.18 

20.25 

45 

46    41.34 

20.17 

41.26 

20.35 

41.17 

20.53 

41.08 

20.70 

46 

47 

42.24 

20.60 

42.15 

20.79 

42.06 

20.97 

41.97 

21.15 

47 

48 

43.14 

21.04 

43.05 

21.23 

42.96 

21.42 

42.86 

21.60 

48 

49 

44.04 

21.48 

43.95 

21.67 

43.85 

21.86 

43.76 

22.05 

49 

50 

44.94 

21.92 

44.84 

22.11 

44.75 

22.31 

44.65 

22.50 

50 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

W 

.2 
Q 

64  Deg. 

63|  Deg. 

63i  Deg. 

634  Deg. 

on 

3 

TRAVERSE    TABLE. 


55 


g 

a. 

P 

26  Deg. 

26*  Deg. 

26|  Deg. 

26*  Deg. 

D 

8 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lmt 

Dep. 

i 

51 

45.84 

22.36! 

45.74 

22756" 

45.64 

22.76 

45.54 

22.9V 

51 

52 

46.74 

22.80 

46.64 

23.00 

46.54 

23.20 

46.43 

23.41 

52 

53 

47.64 

23.23 

47.53 

23.44 

47.43 

23.65 

47.33 

23.86 

53 

54 

48.53 

23.67 

48.43 

23.88 

48.33 

24.09 

48  22 

24.31 

54 

55 

49.43 

24.11 

49.33 

24.33 

49.22 

24.54 

49.11 

24.76 

55 

56 

50.33 

24.55 

50.22 

24.77 

50.12 

24.99 

50.01 

25.21 

56 

57 

51.23 

24.99 

51.12 

25.21 

51.01 

25-43 

50.00 

25.66 

57 

58 

52.13 

25.43 

52.02 

25.65 

51.91 

25  88 

51.79 

26.11 

58 

59 

53.03 

25.86 

52.92 

26.09 

52.80 

26.33 

52.69 

26.56 

59 

60 

53.93 

26.30 

53.81 

26.54 

53.70 

26  77 

53.58 

27.01 

60 

61 

54.83 

26.74 

54.71 

26.98 

54.59 

27.22 

54.47 

27.46 

61 

62 

55.73 

27.18 

55.61 

27.42 

55.49 

27.66 

55.36 

27.91 

62 

63 

56.62 

27.62 

56.50 

27.86 

56.33 

28.11 

56.26 

28.36 

63 

64 

57.52 

23.06 

57.40 

28.31 

57.28 

28.56 

57.15 

28.81 

64 

65 

58.42 

28.49 

58.30 

28.75 

58.17 

29.00 

58.04 

29.26 

65 

66 

59.32 

28.93 

59.19 

29.19 

59.07 

29.45 

58.94 

29.71 

66 

67 

60.22 

29.37 

60.09 

29.63 

59.96 

29.90 

59.83 

30.16 

67 

68 

81.12 

29.81 

60.99 

30.08 

60.86 

30.34 

60.72 

30.61 

68 

69 

62.02 

30.25 

61.88 

30.52 

61.75 

30.79 

61.62 

31.06 

69 

70 

62.92 

30.69 

62.78 

30.96 

62.65 

31.23 

62.51 

31.51 

70 

71 

63.81 

31.12 

63.68 

31.40 

63.54 

31.68  jj  63.40 

31.96 

71 

72 

64.71 

31.56 

64.57 

31.84 

64.44 

32.13  1  64.  29 

32.41 

72 

73 

65.61 

32.00 

65.47 

32.29 

65.33 

32.57  1  65.19 

32.86 

73 

74 

66.51 

32.44 

66.37 

32.73 

66.23 

33.02 

66.08 

33.31 

74 

75 

67.41 

32.88 

67.27 

33.17 

67.12 

33.46 

66.97 

33.76 

75 

76 

68.31 

33.32 

68.16 

33.61 

68.01 

33.91 

67.87 

34.21 

76 

77 

69.21 

33.75 

69.06 

34.06 

68.91 

34.36 

68.76 

34.66 

77 

78 

70.11 

34.19 

69.96 

34.50 

69.80 

34.80 

69.65 

35.11 

78 

79 

71.00 

34.63 

70.85 

34.94 

70.70 

35.25 

70.55 

35.56 

79 

80 

71.90 

35.07 

71.75 

3.5.  33 

71.59 

35.70 

71.44 

36.01 

80 

81 

72.80 

35.51 

72.65 

35  .  83 

72.49 

36.14 

72.33 

36.46 

81 

82 

73.70 

35.95 

73.54 

36.27 

73.38 

36.59 

73.22 

36.91 

82 

83 

74.60 

36.38 

74.44 

36.71 

74.28 

37.03 

74.12 

37.36 

83 

84 

75.50 

36.82 

75.34 

37.15 

75.17 

37.48 

75.01 

37.81 

84 

85 

76.40 

37.26 

76.23 

37.59 

76.07 

37.93 

75.90 

38.26 

85 

86 

77.30 

37.70 

77.13 

38.04 

76.96 

38.37 

76.80 

38.71 

86 

87 

78.20 

88.14 

78.03 

38.48 

77.86 

38.82 

77.69 

39.16 

87 

88 

79.09 

38.58 

78.92 

33.92 

78.75 

39.27 

78.58 

39.61 

88 

89 

79.99 

39.01 

79.82 

39.36 

79.65 

39.71 

79.48 

40.06 

89 

90 

80.89 

39.45 

80.72 

39.81 

80.54 

40.16 

80.37 

40.51 

90 

91 

81.79 

39.89 

81.62 

40.25 

81.44 

40.60 

81.26    40.96 

91 

92 

82.69 

40.33 

82.51 

40.69 

82.33 

41.05 

82.15 

41.41 

92 

93 

83.59 

40.77 

83.41 

41.13 

83.23 

41.50 

83.05 

41.86 

93 

94 

84.49 

41.21 

84.31 

41.58 

84.12 

41.94 

83.94 

42.31 

94 

95 

85.39 

41.65 

85.20 

42.02 

85.02 

42.39 

84.83 

42.76 

95 

96  186.28 

42.08 

86.10 

42.46 

85.91 

42.83 

85.73 

43.21 

96 

97 

87.18 

42.52 

87.00 

42.90 

86.81 

43.28 

86.62 

43.66 

97 

98 

88.08 

42.96 

87.89 

43.34 

87.70 

43.73 

87.51 

44.11 

98 

99 

88.98 

43.40 

88.79 

43.79 

88.60 

44.17 

88.40 

44.56 

99 

100 

89.88 

43.84 

89.69 

44.23 

89.49 

44.62 

89.30 

45.01 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

1 

64  Deg. 

63|  Deg. 

63|  Deg. 

63J  Deg. 

1 

TRAVERSE    TABLE. 


o 

55' 

27  Deg. 

274  Deg. 

27|  Deg. 

27|  Deg. 

5 

•jlT 

P 

| 

9 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat, 

Dep. 

P 

1 

0.89 

0.45 

0.89 

"~O6~ 

0.89 

0.46 

0.88 

0.47 

1 

2 

1.78 

0.91 

1.78 

0.92 

1.77 

0.92 

1.77 

0.93 

2 

3 

2.67 

1.36 

2.67 

1.37 

2.66 

1.39 

2.65 

1.40 

3 

4 

3.56 

1.82 

3.56 

1.83 

3.55 

1.85 

3.54 

1.86 

4 

5 

4.45 

2.27 

4.45 

2,29 

4.44 

2.31 

'4.42 

2.33 

5 

6 

5.35 

2.72 

5.33 

2.75 

5.32 

2.77 

5.31 

2.79 

6 

7 

6.24 

3.18 

6.22 

3.21 

6.21 

3.23 

6.19 

3.26 

7 

8 

7.13 

3.63 

7.11 

3.66 

7.10 

3.69 

7.08 

3.72 

8 

9 

8.02 

4.09 

8.00 

4.12 

7.98 

4.16 

7.96 

4.19 

9 

10 

8.91 

4.54 

8.89 

4.58 

8.87 

4.62 

8.85 

4.66 

10 

11 

9.80 

4.99 

9.78 

5.04 

9.76 

5.08 

9.73 

5.12 

11 

12 

10.69 

5.45 

10.67 

5.49 

10.64 

5.54 

10.62 

5.59 

12 

13 

11.58 

5.90 

11.56 

5.95 

11.53 

6.00 

11.50 

6.05 

13 

14 

12.47 

6.36 

12.45 

6.41 

12.42 

6.46 

12.39 

6.52 

14 

15 

13.37 

6.81 

13.34 

6.87 

13.31 

6.93 

13.27 

6.98 

15 

16 

14.26 

7.26 

14.22 

7.33 

14.19 

7.39 

14.16 

7.45 

16 

17 

15.15 

7.72 

15.11 

7.78 

15.08 

7.85 

15.04 

7.92 

17 

18 

16.04 

8.17 

16.00 

8.24 

15.97 

8.31 

15.93 

8.38 

18 

19 

16.93 

8.63 

16.  89^ 

8.70 

16.85 

8.77 

16.81 

8.85 

19 

20 

17.82 

9.08 

17.78 

9.16 

17.74 

9.23 

17.70 

9.31 

20 

21 

18.71 

9.53 

18.67 

9.62 

18.63 

9.70 

18.58 

9.78 

21 

22 

19.  6D 

9.99 

19.56 

10.07 

19.51 

10.16 

19.47 

10.24 

22 

23 

20.49 

10.44 

20.45 

10.53 

20.40 

10.62 

20.35 

10.71 

23 

24 

21.38 

10.90 

21.34 

10.99 

21.29 

11.08 

21.24 

11.17 

24 

25 

22.28 

11.35 

22.23 

11.45 

22.18 

11.54 

22.12 

11.64 

25 

26 

23.17 

11.80 

23  .  1  1 

11.90 

23.06 

12.01 

23.01 

12.11 

26 

27 

24.06 

12.26 

24.00 

12.36 

23.95 

12.47 

23.89 

12.57 

27 

28 

24.95 

12.71 

24.89 

12.82 

24.84 

12.93 

24.78 

13.04 

28 

29 

25.84 

13.17 

25.78 

13.28 

25  .*  2 

13.39 

25.66 

13.50 

29 

30 

26.73 

13.62 

26.67 

13.74 

26.61 

13.85 

26.55 

13.97 

30 

31 

27.62 

14.07 

27.56 

14.19 

27.50 

14.31 

27.43 

14.43 

31 

32 

28.51 

14.53 

28.45 

14.65 

28.38 

14.78 

28.32 

14.90 

32 

33 

29.40 

14.98 

29.34 

15.11 

29.27 

15.24 

29.20 

15.37 

33 

34 

30.29 

15.44 

30.23 

15.57 

30.16 

15.70 

30.09 

15.83 

34 

35 

31.19 

15.89 

31.12 

16.03 

31.05 

16.16 

30.97 

16.30 

35 

36 

32.08 

16.34 

32.00 

16.48 

31,93 

16.62 

31.86 

16.76 

36 

37 

32.97 

16.80 

32.89 

16.94 

32.82 

17.08 

32.74 

17.23 

37 

38 

33.86 

17.25 

33.78 

17.40 

33.71 

17.55 

33.63 

17.69 

38 

39 

34.75 

17.71 

34.67 

17.86 

34.59 

18.01 

34.51 

18.16 

39 

40 

35.64 

18.16 

35  .  56 

18.31 

35.48 

18.47 

35.40 

18.62 

40 

41 

36.53 

18.61 

36.45 

18.77 

36.37 

18.93 

36.28 

19.09 

41 

42 

37.42 

19.07 

37.34 

19.23 

37.25 

19.39 

37.17 

19.56 

42 

43 

38.31 

19.52 

38.23 

19.69 

38.14 

19.86 

38.05 

20.02 

43 

44 

39.20 

19.98 

39.12 

20.15 

39.03 

20.32 

38.94 

20.49 

44 

45 

40.10 

20.43 

40.01 

20.60 

39.92 

20.78 

39.82 

20.95 

45 

46 

40.99 

20.88 

40.89 

21.06 

40.80 

21.24 

40.71 

21.42 

46 

47 

41.88 

21.34 

41.78 

21.52 

41.69 

21.70 

41.59 

21.88 

47 

48 

42.77 

21.79 

42.67 

21.98 

42.58 

22.16 

42.48 

22.35 

48 

49 

43.66 

22.25 

43.56 

22.44 

43.46 

22.63 

43.36 

'22.82 

49 

50 

44.55 

22.70 

44.45 

22.89 

44.35 

23.09 

44.25 

23.28 

50 

I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

Q 

63  Deg. 

62|  Deg. 

62|  Deg. 

62*  Deg. 

5 

TRAVERSE   TABLE. 


o 

5T 

27-Deg. 

274  Deg. 

271  Deg. 

27|  Deg. 

O 

3 
o 
o 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

"61 

45.44 

23.15 

45.34 

23.35 

45.24 

23.55 

45.13 

23.75 

~51 

52 

46.33 

23.61 

46.23 

23.81 

46.12 

24.01 

46.02 

24.21 

52 

53 

47.22 

24.06 

47.12 

24.27 

47.01 

24.47 

46.90 

24.68 

53 

54 

48.11 

24.52 

43.01 

24.73 

47.90 

24.93 

47.79 

25.14 

54 

55 

49.01 

24.97 

48.90 

25.18 

48.79 

25.40 

48.67 

25.61 

55 

56 

49.90 

25.42 

49.78 

25.64 

49.67 

25.86 

49.56 

26.07 

56 

57 

50.79 

25.88 

50.67 

26.10 

50.56 

26.32 

50.44 

26.54 

57 

58 

51.68 

26.33 

51.56 

26.56 

51.45 

26.78 

51.33 

27.01 

58 

59 

52.57 

26.79 

52.45 

27.01 

52.33 

27.24 

52.21 

27.47 

59 

60 

53.46 

27.24 

53.34 

27.47 

53.22 

27.70 

'53.10 

27.94 

60 

61 

54.35 

27.69 

54.23 

27.93 

54.11 

28.17 

53.98 

28.40 

61 

62 

55.24 

28.15 

55.12 

28.39 

54.99 

28.63 

54.87 

28.87 

62 

63 

56.13 

28.60 

56.01 

28.85 

55.88 

29.09 

55.75 

29.33 

63 

64 

57.02 

29.06 

66.90 

29.30 

56.77 

29.55 

56.64 

29.80 

64 

65 

57.92 

29.51 

57.79 

29.76 

57.66 

30.01 

57.52 

30.26 

65 

66 

58.81 

29.96 

58.68 

30.22 

58.54 

30.48 

58.41 

30.73 

66 

67 

59.70 

30.42 

59.56 

30.68 

59.43 

30.94 

59.29 

31.20 

67 

68 

60.59 

30.87 

60.45 

31.14 

60.32 

31.40 

60.18 

31.66 

68 

69 

61.48 

31.33 

61.34 

31.59 

61.20 

31.86 

61.06 

32.13 

69 

70 

62.37 

31.78 

62.23 

32.05 

62.09 

32.32 

61.95 

32.59 

70 

71 

63.26 

32.23 

63.1* 

32.51 

62.98 

32.78 

62.83 

33.06 

71 

72 

64.15 

32.69 

64.01 

32.97 

63.86 

33.25 

63.72 

33.52 

72 

73 

65.04 

33.14 

64.90 

33.42 

64.75 

33.71 

64.60 

33.99 

73 

74 

65.93 

33.60 

65.79 

33.88 

65.64 

34.17 

65.49 

34.46 

74 

75 

66.83 

34.05 

66.68 

34.34 

66.53 

34.63 

66.37 

34.92 

75 

76 

67.72 

34.50 

67.57 

34.80 

67.41 

35.09 

67.26 

35.39 

76 

77 

68.61 

34.96 

68.45 

35.26 

68.30 

35.55 

68.14 

35.85 

77 

78 

69.60 

35.41 

69.34 

35.71 

69.19 

36.02 

69.03 

36.32 

78 

79 

70.39 

35.87 

70.23 

36.17 

70.07 

36.48 

69.91 

36.78 

79 

80 

71.28 

36.32 

71-12 

36.63 

70.96 

36.94 

70.80 

37.25 

80 

81 

72.17 

36.77 

72.01 

37.09 

71.85 

37.40 

71.68 

37.71 

81 

82 

73.06 

37.23 

72.90 

37.55 

72.73 

37.86 

72.57 

38.18 

82 

83 

73.95 

37.68 

73.79 

38.00 

73.62 

38.33 

73.45 

38.65 

83 

84 

74.84 

38.14 

74.68 

38.46 

74.51 

38.79 

74.34 

39.11 

84 

85 

75.74 

38.59 

75.57 

38.92 

75.40 

39.25 

75.22 

39.58 

85 

86 

76.63 

39.04 

76.46 

39.38 

76.28 

39.71 

76.11 

40.04 

86 

87 

77.52 

39.50 

77.34 

39.83 

77.17 

40.17 

76.99 

40.51 

87 

88 

78.41 

39.95 

78.23 

40.29 

78.06 

40.63 

77.88 

40.97 

88 

89 

79.30 

40.41 

79.12 

40.75 

78.94 

41.10 

78.76 

41.44 

89 

90 

80.19 

40.86 

80.01 

41.21 

79.83 

41.56 

79.65 

41.91 

90 

91 

81.08 

41.31 

80.90 

41.67 

80.72 

42.02 

80.53 

42.37 

91 

92 

81.97 

41.77 

81.79 

42.12 

81.60 

42.48 

81.42 

42.84 

92 

93 

82.86 

42.22 

82.63 

42.58 

82.49 

42.94 

82.30 

43.30 

93 

94 

83.75 

42.68 

83.57 

43.04 

83.38 

43.40 

83.19 

43.77 

94 

95 

84.65 

43.13 

84.46 

43.50 

84.27 

43.87 

84.07 

44.23 

95 

96 

85.54 

43.58 

85.35 

43.96 

85.15 

44.33 

84.96 

44.70 

96 

97 

86.43 

44.04 

86.23 

44.41 

86.04 

44.79 

85.84 

45.16 

97 

98 

87.32 

44.49 

87.12 

44.87 

86.93 

45.25 

86.73 

45.63 

98 

99 

88.21 

44.95 

88.01 

45.33 

87.81 

45.71 

87.61 

46.10 

99 

100 

89.10 

45.40 

88.90 

45.79 

88.70 

46.17 

88.50 

46.56 

100 

T 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

J3 

1 

g 

63  Deg. 

62|  Deg. 

62£  Deg. 

62}  Deg. 

s 

TRAVERSE    TABLE. 


] 

0 

28  Deg. 

284  Deg. 

28f  Deg. 

28|  Deg! 

t? 

£* 

? 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

1 

0.88 

0.47 

0.88 

0.47 

0.88 

0.48 

0.88 

0.48 

1 

2 

1.77 

0.94 

1.76 

0.95 

1.76 

0.95 

1.75 

0.96 

2 

3 

2.65 

1.41 

2.64 

1.42 

2.64 

1.43 

2.63 

1.44 

3 

4 

3.53 

1.88 

3.52 

1.89 

3.52 

1.91 

3.51 

1.92 

4 

5 

4.41 

2.35 

4.40 

2.37 

4.39 

2.39 

4.38 

2.40 

5 

6 

5.30 

2.82 

5.29 

2.84 

5.27 

2.86 

5.26 

2.89 

6 

7 

6.18 

3.29 

6.17 

3.31 

6.15 

3.34 

6.14 

3.37 

7 

8 

7.06 

3.76 

7.05 

3.79 

7.03 

3.82 

7.01 

3.85 

8 

9 

7:95 

4.23 

7.93 

4.26 

7.91 

4.29 

7.89 

4.33 

9 

10 

8.83 

4.69 

8.81 

4.73 

8.79 

4.77 

8.77 

4.81 

10 

11 

9.71 

5.16 

9.69 

5.21 

9.67 

5.25 

9.64 

5.29 

11 

12 

10.60 

5.63 

10.57 

5.68 

10.55 

5.73 

JO.  52 

5.77 

12 

13 

11.48 

6.10 

11.45 

6.15 

11.42 

6.20 

11.40 

6.25 

13 

14 

12.36 

6.57 

12.33 

6.63 

12.30 

6.68 

12.27 

6.73 

14 

15 

13.24 

7.04 

13.21 

7.10 

13.18 

7.16 

13.15 

7.21 

15 

16 

14.13 

7.51 

14.09 

7.57 

14.06 

7.63 

14.03 

7.70 

16 

17 

15.01 

7.98 

14.98 

8.05 

14.94 

8.11 

14.90 

8.18 

17 

18 

15.89 

8.45 

15.86 

8.52 

15.82 

8.59 

15.78 

8.66 

18 

19 

16.78 

8.92 

16.74 

8.99 

16.70 

9.07 

16,66 

9.14 

19 

20 

17.66 

9.39 

17.62 

9.47 

17.58 

ft}9'54 

17.53 

9.62 

20 

21 

18.54 

9.86 

18.50 

9.94 

18.46 

TO.  02 

18.41 

10.10 

21 

22 

19.42 

10.33 

19.38 

10.41 

19.33 

10.50 

19.29 

10.58 

22 

23 

20.31 

10.80 

20.26 

10.89 

20.21 

10.97 

20.16 

11.06 

23 

24 

21.19 

11.27 

21.14 

11.36 

21.09 

11.45 

21.04 

11.54 

24 

25 

22.07 

11.74 

22.02 

11.83 

21.97 

11.931 

21.92 

12.02 

25 

26 

22.96 

12.21 

22.90 

12,31 

22.85 

12.41 

22.79 

12.51 

26 

27 

23.84 

12.68 

23  .  78 

12.78 

23.73 

12.88 

23.67 

12.99 

27 

28 

24.72 

13.15 

24.66 

13.25 

24.61 

13.36 

24.55 

13.47 

28 

29 

25.61 

13.61 

25.55 

13.73 

25.49 

13.84 

25.43 

13.95 

29 

30 

26.49 

14.08 

26.43 

14.20 

26.36 

14.31 

26.30 

14.43 

30 

31 

27.37 

14.55 

27.31 

14.67 

27.24 

14.79 

27.18 

14.91 

31 

32 

28.25 

15.02 

28.19 

15.15 

28.12 

15.27 

28.06 

15.39 

32 

33 

29  .44 

15.49 

29.07 

15.62 

29.00 

15.75 

28.93 

15.87 

33 

34 

30.02 

15.96 

29.95 

16.09 

29.88 

16.22 

29.81 

16.35 

34 

35 

30.90 

16.43 

30.83 

16.57 

30.76 

16.70 

30.69 

16.83 

35 

36 

31.79 

16.90 

31.71 

17.04 

31.64 

17.18 

31.56 

17.32 

36 

37 

32.67 

17.37 

32.59 

17.51 

32.52 

17.65 

32.44 

17.80 

37 

38 

33.55 

17.84 

33.47 

17.99 

33.39 

18.13 

33.32 

18.28 

38 

39 

34.43 

18.31 

34.35 

18.46 

34.27 

18.61 

34.19 

18.76 

39 

40 

35.32 

18.78 

35.24 

18.93 

35.15 

19.09 

35.07 

19.24 

40 

41 

36.20 

19.25 

36.12 

19.41 

36.03 

19.56 

35.95 

19.72 

41 

42 

37.08 

19.72 

37.00 

19.88 

36.91 

20.04 

36.82 

20.20 

42 

43 

37.97 

20.19 

37.88 

20.35 

37.79 

20.52 

37.70 

20.68 

43 

44 

38.85 

20.66 

38.76 

20.83 

38.67 

20.99 

38.58 

21.16 

44 

45 

39.73 

21.13 

39.64 

21.30 

39.55 

21.47 

39.45 

21.64 

45 

46 

40.62 

21.60 

40.52 

21.77 

40.43 

21.95 

40.33 

22.13 

46 

47 

41.50 

22.07 

41.40 

22.25 

41.30 

22.43 

41.21 

22.61 

47 

48 

42.38 

22.53 

42.28 

22.72 

42.18 

22.90 

42.08 

23.09 

48 

49 

43.26 

23.00 

43.16 

23.19 

43.06 

23.38 

42.96 

23.57 

49 

50 

44.15 

23.47 

44.04 

23.67 

43.94 

23.86 

43.84 

24.05 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

0 

to 

p 

62  Deg. 

61|  Deg. 

61^  Deg. 

6H  Deg. 

Q 

r 

TRAVERSE    TABLE. 


59 


G 

28  Deg. 

28i  Deg. 

28iDeg. 

28!  Deg. 

d 
35' 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o 
? 

51 

45.03 

23.94 

44.93 

24.14 

44.82 

24.34j 

44.71 

24.53 

~5~1 

52 

45.91 

24.41 

45.81 

24.61 

45.70 

24,81 

45.59 

25.01 

52 

53 

46.80 

24.88 

46.69 

25.09 

46.58 

25.29 

46.47 

25.49 

53 

54 

47.68 

25.35 

47.57 

25.56 

47.46 

25.77 

47.34 

25.97 

54 

55 

48.56 

25.82 

48.45 

26.03 

48.33 

26.24 

48.22 

26.45 

55 

56 

49.45 

26.29 

49.33 

26.51 

49.21 

26.72 

49.10 

26.94 

56 

57 

50.33 

26.76 

50.21 

26.98 

50.09 

27.20 

49.97 

27.42 

57 

58 

51.21 

27.23 

51.09 

27.45 

50.97 

27.68 

50.85 

27.90 

58 

59 

52.09 

27.70 

51.97 

27.93 

51.85 

28.15 

51.73 

28.38 

59 

60 

52.98 

28.17 

52.85 

28,40 

52.73 

28.63 

52.60 

28,86 

60 

61 

53.86 

28.64 

53.73 

28.87 

53.61 

29.11 

53.48 

29.34 

61 

62 

54.74 

29.11 

54.62 

29.35 

54.49 

29.58 

54.36 

29.82 

62 

63 

55.63 

29.58 

55.50 

29.82 

55.37 

30.06 

55.23 

30.30 

63 

64 

56.51 

30.05 

56.38 

30.29 

56.24 

30.54 

56.11 

30.78 

64 

65 

57.39 

30.52 

57.26 

30.77 

57.12 

31.02 

56.99 

31.26 

65 

66 

58.27 

30.99 

58.14 

31.24 

58.00 

31.49 

57.86 

31.75 

66 

67 

59.16 

31.45 

59.02 

31.71 

58.88 

31.97 

58.74 

32.23 

67 

68 

60.04 

31.92 

59.90 

32.19 

59.76 

32.45 

59.62 

32.71 

68 

69 

60.92 

32.39 

60.78 

32.66 

60.64 

32.92 

60.49 

33.19 

69 

70 

61.81 

32.86 

61.66 

33.13 

61.52 

33.40 

61.37 

33.67 

70 

71 

62.69 

33.33 

62.54 

33.61 

62.40 

33.88 

62.25 

34.15 

71 

72 

63.57 

33.80 

63.42 

34.08 

;63.27 

34.36 

63.12 

34.63 

72 

73 

64.46 

34.27 

64.30 

34.55 

164.15 

34.83 

64.00 

35.11 

73 

74 

65.34 

34.74 

65.19 

35.03 

65.03 

35.31 

'64.88 

35.59 

74 

75 

66.22 

35.21 

66.07 

35.50 

65.91 

35.79 

65.75 

36.07 

75 

76 

67.10 

35.68 

66.95 

35.97 

66.79 

36.26 

,66.63 

36.56 

76 

77 

67.99 

36.15 

67.83 

36.45 

67.67 

36.74 

167.51 

37.04 

77 

78 

68.87 

36.62 

68.71 

36.92 

68.55 

37.22 

68.38 

37.52 

78 

79 

69.75 

37.09 

69.59 

37.39 

69.43 

37.70 

!69.26 

38.00 

79 

80 

70:64 

37,56 

70.47 

37.87 

70.31 

38.17 

170.14 

38.48 

80 

81 

71.52 

38.03 

»71.35 

38.34 

71.18 

38.65 

71  .01 

38.96 

81 

82 

72.40 

38.50 

72.23 

38.81 

72.06 

39.13 

71.89 

39.44 

82 

83 

73.28 

38.97 

73.11 

39.29 

72.94 

39.60 

J72.77 

39.92 

83 

84 

74.17 

39.44 

73.99 

39.76 

73.82 

40.08 

73.64 

40.40 

84 

85 

75.05 

39.91 

74.88 

40.23 

74.70 

40.56 

74.52 

40.88 

85 

86 

75  .  93 

40.37 

75.76 

40.71 

75  58 

41.04 

75.40 

41.36 

86 

87 

76.82 

40.84 

76.64 

41.18 

76.46 

41.51 

76.28 

41.85 

87 

88 

77.70 

41.31 

77.52 

41.65 

77.34 

41.99 

77.15 

42.33 

88 

89 

78.58 

41.78 

78.40 

42.13 

78.21 

42.47 

78.03 

42.81 

89 

90 

79.47 

42.25 

79.28 

42.60 

79.09 

42.94 

78.91 

43.29 

90 

91 

80.35 

42.72 

80.16 

43.07 

79.97 

43.42 

79.78 

43.77 

91 

92 

81.23 

43.19 

81.04 

43.55 

80.85 

43.90 

80.66 

44.25 

92 

93 

82.11 

43.66 

81.92 

44.02 

81.73 

44.38 

81.54 

44.73 

93 

94 

83.00 

44.13 

82.80 

44.49 

82.61 

44.85 

82.41 

45.21 

94 

95 

83.88 

44.60 

83.68 

44.97 

83.49 

45.33 

83.29 

45.69 

95 

96 

84.76 

45.07 

84.57 

45.44 

84.37 

45.81 

84.17 

46.17 

96 

97 

85.65 

45.54 

85.45 

45.91 

85.25 

46.28 

J85.04 

46.66 

97 

98 

86.53 

46.01 

86.33 

46.39 

86.12 

46.76 

85.92 

47.14 

98 

99 

87.41 

46.48 

87.21 

46.86 

87.00 

47.24 

86.80 

47.62 

99 

100 

88.29 

46.95 

88.09 

47.33 

87.88 

47.72 

87.67 

48.10 

100 

8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

6 

Q 

62  Deg. 

61}  Deg. 

61i  Deg. 

6H  Deg. 

s 

60 


TRAVERSE    TABLE. 


o 

5T 

29  Deg. 

294  Deg. 

29i  Deg. 

29|  Deg. 

C 
5' 

Jtf 

9> 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

V 
o 

0> 

1 

0.87 

0.48 

0.87 

0.49 

0.87 

0.49  1 

0.87 

0.50 

1~ 

2 

1.75 

0.97 

1.74 

0.98 

1.74 

0.98 

1.74 

0.99 

2 

3 

2.62 

1.45 

2.62 

1.47 

2.61 

1.48 

2.60 

1.49 

3 

4 

3.50 

1.94 

3.49 

1.95 

3.48 

1.97 

3.47 

1.98 

4 

5 

4.37 

2.42 

4.36 

2.44 

4.35 

2.46 

4.34 

2.48 

5 

6 

5.25 

2.91 

5.23 

2.93 

5.22 

2.95 

5.21 

2.98 

6 

7 

6.12 

3.39 

6.11 

3.42 

6.09 

3.45 

6.08 

3.47 

7 

8 

7.00 

3.88 

6.98 

3.91 

6.96 

3.94 

6.95 

3.97 

8 

9 

7.87 

4.36 

7.85 

4.40 

7.83 

4.43 

7.81 

4.47 

9 

10 

8.75 

4.85 

8.72 

4.89 

8.70 

4.92 

8.68 

4.96 

10 

11 

9.62 

5.33 

9.60 

5.37 

9.57 

5.42 

9.55 

5.46 

11 

12 

10.50 

5.82 

10.47 

5.86 

10.44 

5.91 

10.42 

5.95 

12 

13 

11.37 

6.30 

11.34 

6.35 

11.31 

6.40 

11.29 

6.45 

13 

14 

12.24 

6.79 

12.21 

6.84 

12.18 

6.89 

12.15 

6.95 

14 

15 

13.12 

7.27 

13.09 

7.33 

13.06 

7.39 

13.02 

7.44 

15 

16 

13.99 

7.76 

13.96 

7.8t 

13.93 

7.88 

13.89 

7.94 

16 

17 

14.87 

8.24 

14.83 

8.31 

14.80 

8.37 

14.76 

8.44 

17 

18 

15.74 

8.73 

15.70 

8.80 

15.67 

8.86 

15.63 

8.93 

18 

19 

16.62 

9.21 

16.58 

9.28 

16.54 

9.36 

16.50 

9.43 

19 

20 

17.49 

9.70 

17.45 

9.77 

17.41 

9.85 

17.36 

9.92 

20 

21 

18.31 

10.18 

18.32 

10.26 

18.28 

10.34 

18.23 

10.42 

21 

22 

19.24 

10.67 

19.19 

10.75 

19.15 

10.83 

19.10 

10.92 

22 

23 

20.12 

11.15 

20.07 

11.24 

20.02 

11.33 

19.97 

11.41 

23 

24 

20.99 

11.64 

20.94 

11.73 

20.89 

11.82 

20.84 

11.91 

24 

25 

21.87 

12.12 

21.81 

12.22 

21.76 

12.31 

21.70 

12.41 

25 

26 

22.74 

12.60 

22.68 

12.70 

22.63 

12.80 

22.57 

12.90 

26 

27 

23.61 

13.09 

23.56 

13.19 

23.50 

13.30 

23.44 

13.40 

27 

28 

24.49 

13.57 

24.43 

13.68 

24/37 

13.79 

24.31 

13.89 

28 

29 

25.36 

14.06 

25.30 

14.17 

25.24 

14.28 

25.18 

14.39 

29 

30 

26.24 

14.54 

26.17 

14.66 

26.11 

14.77 

26.05 

14.89 

30 

31 

27.11 

15.03 

27.05 

15.15 

26.98 

15.27 

26.91 

15.38 

31 

32 

27.99 

15.51 

27.92 

15.64 

27.85 

15.76 

27.78 

15.88 

32 

33 

28.86 

16.00 

28.79 

16.12 

28.72 

16.25 

28.65 

16.38 

33 

34 

29.74 

16.48 

29.66 

16.61 

29.59 

16.74 

29.52 

16.87 

34 

35 

30.61 

16.97 

30.54 

17.10 

30.46 

17.23 

30.39 

17.37 

35 

36 

31.49 

17.45 

31.41 

17.59 

31.33 

17.73 

31.26 

17.86 

36 

37 

32.36 

17.94 

32.28 

18.08 

32.20 

18.22 

32.12 

18.36 

37 

38 

33.24 

18.42 

33.15 

18.57 

33.07 

18.71 

32.99 

18.86 

38 

39 

34.11 

18.91 

34.03 

19.06 

33.94 

19.20 

33.86 

19.35 

39 

40 

34.98 

19.39 

34.90 

19.54 

34.81 

19.70 

34.73 

19.85 

40 

41 

35.86 

19.88 

35  .  77 

20.03 

35.68 

20.19 

35.60 

20.34 

41 

42 

36.73 

20.36 

36.64 

20.52 

36.55 

20.68 

36.46 

20.84 

42 

43 

37.61 

20.85 

37.52 

21.01 

37.43 

21.17 

37.33 

21.34 

43 

44 

38.48 

21.33 

38.39 

21.50 

38.30 

21.67 

38.20 

21.83 

44 

45 

39.36 

21.82 

39.26 

21.99 

39.17 

22.16' 

39.07 

22.33 

45 

46 

40.23 

22.30 

40.13 

22.48 

40.04 

22.65 

39.94 

22.83 

46 

47 

41.11 

22.79 

41.01 

22.97 

40.91 

23.14 

40.81 

23.32 

47 

48 

41.98 

23.27 

41.88 

23.45 

41.78 

23.63 

41.67 

23.82 

48 

49 

42.86 

23.76 

42.75 

23.94 

42.65 

24.13 

42.54 

24.31 

49 

50 

43.73 

24.24 

43.62 

24.43 

43.52 

24.62 

43.41 

24.81 

50 

| 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

3 

61  Deg. 

1 

60|  Deg. 

60i  Deg. 

60J  Deg. 

ri 
y; 

3 

TRAVERSE   TABLE, 


Q 

29  Deg. 

29*  Deg. 

29£  Deg. 

29}  Deg. 

s 

P 

1 

f5 
CO 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

51 

44.61 

24.73 

44.50 

24.92 

44.39 

25.11 

44.28 

25.31 

51 

52 

45.48 

25.21 

45.37 

^5.41 

45.26 

25.61 

45.15 

25.80 

52 

53 

46.35 

25.69 

46.24 

25.90 

46.13 

26.10 

46.01 

26.30 

53 

54 

47.23 

26.18 

47.11 

26.39 

47.00 

26.59 

46.88 

26,80 

54 

55 

48.10 

26.66 

47.99 

26,87 

47.87 

27.08 

47.75 

27.29 

55 

56 

48.98 

27.15 

48.86 

27.36 

48.74 

27.58 

48.62 

27.79 

56 

57 

49.85 

27.63 

49.73 

27.  8£ 

49.61 

28.07 

49.49 

28.28 

57 

58 

50.73 

28.12 

50.60 

28'.34 

50.48 

28V56 

50.36 

28.78 

58 

59 

51.60 

28.60 

51.48 

28.83 

51.35 

29.05 

51.22 

29.28 

59 

60 

52.48 

29.09 

52.35 

29.32 

52.22 

29.55 

52.09 

29.77 

60 

61 

53.35 

29.57 

53.  *2 

29.81 

53.09 

30.04 

52.96 

30.27. 

61 

62 

54.23 

30.06 

54.09 

30.29 

53.96 

30.53 

53.83 

30.77 

62 

63 

55.10 

30.54 

54.97 

30.78 

54.83 

31.02 

54.70 

31.26 

63 

64 

55.98 

31.03 

55.84 

31.27 

55.70 

31.52 

55.56 

31.76 

64 

65 

56.85 

31.5-1 

56.71 

31.76 

56.57 

32.01 

56.43 

32.25 

65 

66 

57.72 

32.00 

57.58 

32.25 

57.44 

32.50 

57.30 

32.75 

66 

67 

58.60 

32.48 

58.46 

32.74 

58.31 

32.99 

58.17 

33.25 

67 

68 

59.47 

32.97 

59.33 

33.23 

59.18 

33.48 

59.04 

33.74 

68 

69 

60.35 

33.45 

60.20 

33.71 

60.05 

33.98 

59.91 

34.24 

69 

70 

61.22 

33.94 

61.07 

34.20 

60.92 

34.47 

60.77 

34.74 

70 

71 

62.10 

34.42 

61.95 

34.69 

61.80 

34.96 

61.64 

35.23 

71 

72 

62.97 

34.91 

62.82 

35.18 

62.67 

35.45 

62.51 

35.73 

72 

73 

63.85 

35.39 

63.69 

35.67 

63.54 

35.95 

63.38 

36.22 

73 

74 

64.72 

35.88 

64.56 

36.16 

64.41 

36.44 

64.25 

36.72 

74 

75 

65.60 

36.36 

65.44 

36.65 

65.28 

36.93 

65.11 

37.22 

75 

76 

66.47 

36.85 

66.31 

37.14 

66.15 

37.42 

65.98 

37.71 

76 

Z7 

67.35 

37.33 

67.18 

37.62 

67.02 

37.92 

66.80 

38.21 

77 

78 

68.22 

37.82 

68.05 

38.11 

67.89 

38.41 

67.72 

38.70 

78 

79 

69.09 

38.30 

68.93 

38.60 

68.76 

38.90 

68.59 

39.20 

79 

80 

69.97 

38.78 

69.80 

39.09 

69.63 

39.39 

69.46 

39.70 

80 

81 

70.84 

39.27 

70.67 

39.58 

70.50 

39.89 

70.32 

40.19 

81 

82 

71.72 

39.75 

71.54 

40.07 

71.37 

40.38 

71.19 

40.69 

82 

83 

72.59 

40.24 

72.42 

40.56 

72.24 

40.87 

72.06 

41.19 

83 

84 

73.47 

40.72 

73.29 

41.04 

73.11 

41.36 

72.93 

41.68 

84 

85 

74.34 

41.21 

74.16 

41.53 

73.98 

41.86 

73.80 

42.18 

85 

86 

75.22 

41.69 

75.03 

42.02 

74.85 

42.35 

74.67 

42.67 

86 

87 

76.09 

42.18 

75.91 

42.51 

75.72 

42.84 

75.53 

43.17     87 

88 

76.97 

42.65 

76.78 

43.00 

76.59 

43.33 

76.40 

43.67 

88 

89 

77.  §4 

43.15 

77.65 

43.49 

77.46 

43.83 

77.27 

44.16 

89 

90 

78.72 

43.63 

78.52 

43.98 

78,33 

44.32 

78.14 

44.66 

90 

91 

79.59 

44.12 

79.40 

44.46 

79.20 

44.81 

79.01 

45.16 

91 

92 

80.46 

44.60 

80.27 

44.95 

80.07 

45.30 

79.87 

45.65 

92 

93 

81.34 

45.09 

81.14 

45.44 

80.94 

45.80 

80.74 

46.15 

93 

94 

82.21 

45.57 

82.01 

45.93 

81.81 

46.29 

81.61 

46.6-1 

94 

95 

83.09 

46.06 

82.89 

46.42 

82.68 

46.78 

82.48 

47.14 

95 

96 

83.96 

46.54 

83.76 

46.91 

83.55 

47.27 

83.35 

47.64 

96 

97 

84.84 

47.03 

84.63 

47.40 

84.42 

47.77 

84.22 

48.13 

97 

98 

85.71 

47.51 

85.50 

47.88 

85.  2B 

48.26 

85.08 

48.63 

98 

99 

86.59 

48.00 

86.38 

48.37 

86.17 

48.75 

85.95 

49.13 

99 

100 

87.46    48.48 

87.25 

48,86 

87.04 

49.24 

86.82 

49.62 

100 

§ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o> 
o 

c 

5 

.2 

s 

61  Deg. 

60|  Deg. 

60i  Deg. 

60*  Deg. 

3 

62 


TRAVERSE    TABLE, 


o 

30  Deg. 

30i  Deg. 

30£  Deg. 

30|  Deg. 

O 

CO 

5' 
P 

I 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

1 

0.87 

0.50 

0.86 

0.50 

0.86 

0.51 

0.86 

0.51 

i 

2 

1.73 

1.00 

1.73 

1.01 

1.72 

1.02 

1.72 

1.02 

2 

3 

2.60 

1.50 

2.59 

1.51 

2.58 

1,52 

2.58 

1.53 

3 

4 

3.46 

2.00 

3.46 

2.02 

3.45 

2.03 

3.44 

2.05 

4 

5 

4.33 

2.50 

4.32 

2.52 

4.31 

2.54 

4.30 

2.56 

5 

6 

5.20 

3.00 

5.18 

3.02 

5.17 

3.05 

5.16 

3.07 

6 

7 

6.06 

3.50 

6.05 

3.53 

6.03 

3.55 

6.02 

3.58 

7 

8 

6.93 

4.00 

6.91 

4.03 

6.89 

4.06 

6.88 

4.09 

8 

9 

7.79 

4.50 

7.77 

4.53 

7.75 

4.57 

7.73 

4.60 

9 

10 

8.66 

5.00 

8.64 

5.04 

8.62 

5.08 

8.59 

5.11 

10 

11 

9.53 

5.50 

9.50 

5.54 

9.48 

5.58 

9.45 

5.62 

11 

12 

10.39 

6.00 

10.37 

6.05 

10.34 

6.09 

10.31 

6.14 

12 

13 

11.26 

6.50 

11.23 

6.55 

11.20 

6.60 

11.17 

6  -.65 

13 

14 

12.12 

7.00 

12.09 

7.05 

12.06 

7.11 

12.03 

7.16 

14 

15 

12.99 

7.50 

12.96 

7.56 

12.92 

7.61 

12.89 

7.67 

15 

16 

13.86 

8.00 

13.82 

8.06 

13.79 

8.12 

13.75 

8.18 

16 

17 

14.72 

8.50 

14.69 

8.56 

14.65 

8.63 

14.61 

8.69 

17 

18 

15.59 

9.00 

15.55 

9.07 

15.51 

9.14 

15.47 

9.20 

18 

19 

16.45 

9.50 

16.41 

9.57 

16.37 

9.64 

16.33 

9.71 

19 

20 

17.32 

10.00 

17.28 

10.08 

17.23 

10.15 

17.19 

10.23 

20 

21 

18.19 

10.50 

18.14 

10.58 

18.09 

10.66 

18.05 

10.74 

21 

22 

19.05 

11.00 

19.00 

11.08 

18.96 

11.17 

18.91 

11.25 

22 

23 

19.92 

11.50 

19.87 

11.59 

19.82 

11.67 

19.77 

11.76 

23 

24 

20.78 

12.00 

20.73 

12.09 

20  .  68 

12.18 

20.63 

12.27 

24 

25 

21.65 

12.50 

21.60 

12.59 

21.54 

12.69 

21.49 

12.78 

25 

26 

22.52 

13.00 

22.46 

13.10 

22.40 

13.20 

22.34 

13.29 

26 

27 

23.38 

13.50 

23.32 

13.60 

23.26 

13.70 

23.20 

13.80 

27 

28 

24.25 

14.00 

24.19 

14.11 

24.13 

14.21 

24.06 

14.32 

28 

29 

25.11 

14.50 

25.05 

14.61 

24.99 

14.72 

24.92 

14.83 

29 

30 

29.98 

15.00 

25.92 

15.11 

25.85 

15.23 

25.78 

15.34 

30 

31 

26.85 

15.50 

26.78 

15.62 

26.71 

15.73 

26.64 

15.85 

31 

32 

27.71 

16.00 

27.64 

16.12 

27.57 

16.24 

27.50 

16.36 

32 

33 

28  .  58 

16.50 

28.51 

16.62 

28.43 

16.75 

28.36 

16.87 

33 

34 

29.44 

17.00 

29.37 

17.13 

29.30 

17.26 

29.22 

17.38 

34 

35 

30.31 

17.50 

30.23 

17.63 

30.16 

17.76 

30.08 

17.90 

35 

36 

31.18 

18.00 

31.10 

18.14 

31.02 

18.27 

30.94 

18.41 

36 

37 

32.04 

18.50 

31.96 

18.64 

31.88 

18.78 

31.80 

18.92 

37 

38 

32.91 

19.00 

32.83 

19.14 

32.74 

19.29 

32.66 

19.43 

38 

39 

33.77 

19.50 

33.69 

19.65 

33.60 

19.79 

33.52 

19.94 

39 

40 

34.64 

20.00 

34.55 

20.15 

34.47 

20.30 

34.38 

20.45 

40 

41 

35.51 

20.50 

35.42 

20.65 

35.33 

20.81 

35.24 

20.96 

41 

42 

36.37 

21.00 

36.28 

21.16 

36.19 

21.32 

36.10 

21.47 

42 

43 

37.24 

21.50 

37.14 

21.66 

37.05 

21.82 

36.95 

21.99 

43 

44 

38.11 

22.00 

38.01 

22.17 

37.91 

22.33 

37.81 

22.50 

44 

45 

38.97 

22.50 

38.87 

22.67 

38  .  77 

22.84 

38.67 

23.01 

45 

46 

39.84 

23.00 

39.74 

23.17 

39.63 

23.35 

39.53 

23.52 

46 

47 

40.70 

23.50 

40.60 

23.68 

40.50 

23.85 

40.39 

24.03 

47 

48 

41.57 

24.00 

41.46 

24.18 

41.36 

24.36 

41  .'25 

24.54 

48 

49 

42.44 

24.50 

42.33 

24.68 

42.22 

24.87 

42.11 

25.05 

49 

50 

43.30 

25.00 

43.19 

25.19 

43.08 

25.38 

42.97 

25.56 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

a! 
o 
a 

.2 

q 

60  Deg. 

59|  Dog. 

59|  Deg. 

'  59i  Deg. 

£ 

"to 

Q 

TRAVERSE    TABLE. 


g 

30  Deg. 

30$  Deg. 

30*  Deg. 

30,  Deg. 

O 

QQ 

S 

1' 

n 
o 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

I 

~51 

44.17 

25.50 

44.06 

25.69 

43.94 

25.88 

43.83 

26.08 

51 

52 

45.03 

26.00 

44.92 

26.20 

44.80 

26.39 

44.69 

26.59 

52 

53 

45.90 

26.50 

45.78 

26  70 

45.67 

26.90 

45.55 

27.10 

53 

54 

46.77 

27.00 

46.65 

27.20 

46.53 

27.41 

46.41 

27.61 

54 

55 

47.63 

27.50 

47.51 

27.71 

47.39 

27.91 

47.27 

28.12 

55 

56 

48.50 

28.00 

48.37 

28.21 

48.25 

28.42 

43.13 

28.63 

56 

57 

49.36 

28.50 

4-9.24 

28.72 

49.11 

28.93 

48.99 

29.14 

57 

58 

50.23 

29.00 

50.10 

29.22 

49.97 

29.44 

49.85 

29.65 

58 

59 

51.10 

29.50 

50.97 

29.72 

50.84 

29.94 

50.70 

30.17 

59 

60 

51.96 

3U.OO 

51.83 

30.23 

51.70 

30.45 

51.56 

30.68 

60 

61 

52.83 

30.50 

52.69 

30.73 

52.56 

30.96 

52.42 

31.19 

61 

62 

53.69 

31.00 

53.56 

31.23 

53.42 

31.47 

53.28 

31.70 

62 

63 

54.56 

31.50 

54.42 

31.74 

54.28 

31.97 

54.14 

32.21 

63 

64 

55.43 

32.00 

55.29 

32.24 

55.14 

32.48 

55.00 

32.72 

04 

65 

56.29 

32.50 

56.15 

32.75 

56.01 

32.99 

55.86 

33.23 

65 

66 

57.16 

33.00 

57.01 

33.25 

56.87 

33.50 

56  .  72 

33.75 

66 

67 

58.02 

33.50 

57.88 

33.75 

07.73 

34.01 

57.58 

34.26 

67 

68 

5S.89 

34.00 

58.74 

34.26 

58.59 

34.51 

58.44 

34.77 

68 

*  69 

59.76 

34.50 

59.60 

34.76 

59.45 

35.02 

59.30 

35.28 

69 

70 

60.62 

35.00 

60.47 

35.26 

60.31 

35.53 

60.16 

35.79 

70 

71 

61.49 

35.50  1 

61.33 

35.77 

61.18 

36.04 

61.02 

36.30 

71 

72 

62.35 

36.00 

62.20 

36.27 

62.04 

36.54 

61.88 

36.81 

72 

73 

63.22 

36.50 

63.06 

36.78 

62.90 

37.05 

62.74 

37.32 

73 

74 

64.09 

37.00 

63.92 

37.28 

63.76 

37.56 

63.60 

37.84 

74 

75 

64.95 

37.50 

64.79 

37.78 

64.62 

38.07 

64.46 

38.35 

75 

76 

65.82 

38.00 

65.65 

38.29 

65.48 

38.57 

65.31 

38.86 

76 

77 

66.68 

38.50 

66.52 

38  .  79 

66.35 

39.08 

66.17 

39.37 

77 

78 

67.55 

39.00 

67.38 

39.29 

67.21 

39.59 

67.03 

39.88 

78 

79 

68.42 

39.50. 

68.24 

39.80 

68.07 

40.10 

67.89 

40.39 

79 

80 

69.28 

40.00 

69.11 

40.30 

68.93 

40.60 

68.75 

40.90 

80 

81 

70.15 

40.50 

69.97" 

40.81 

69.79 

41.11 

69.61 

41.41 

81 

82 

71.01 

41.00 

70.83 

41.31 

70.65 

41.62 

70.47 

41.93 

82 

83 

71.88 

41.50 

71.70 

41.81 

71.52 

42.13 

71.33 

42.44 

83 

84 

72.75 

42.00 

72.56 

42.32 

72.38 

42.63 

72.19 

42.95 

84 

85 

73.61 

42.50 

73.43 

42.82 

73.24 

43.14 

73.05 

43.46 

85 

86 

74.48 

43.00 

74.29 

43*.  32 

74.10 

43.65 

73.91 

43.97 

86 

87 

75.34 

43.50 

75.15 

43.83 

74.96 

44.16 

74.77 

44.48 

87 

88 

76.21 

44.00 

76.02 

44.33 

75.82 

44.66 

75.63 

44.  S9 

88 

89 

77.08 

44.50 

76.88 

44.84 

76.68 

45.17 

76.49 

45.51 

89 

90 

77.94 

45.00 

77.75 

45.34 

77.55 

45.68 

77.35 

46.02 

90 

91 

78.81 

45.50 

78.61 

45.84 

78.41 

46.19 

78.21 

46.53 

91 

92 

79.67 

46.00 

79.47 

46.35 

79.27 

46.69 

79.07 

47.04 

92 

93 

80.54 

46.50 

80.34 

46.85 

80.13 

47.20 

79.92 

47.55 

93 

94 

81.41 

47.00 

81.20 

47.35 

80.99 

47.71 

80.78 

48.06 

94 

95 

82.27 

47.50 

82.06 

47.86     81.  85  j  48.  22 

81.64 

48.57 

95 

96 

83.14 

48.00 

82.93 

48.36     82.72 

48.72 

82.50 

49.08 

96 

97 

84.00 

48.50 

83.79 

48.87    83.58 

49.23 

83.36 

49.60 

97 

98 

84.87 

49.00 

84.66 

49.37    84.44 

49.74 

84.22 

50.11 

98 

99 

85.74 

49.50 

85.52 

49.87    85.30 

50.25 

85.08 

50.62 

99 

100 

86.60 

50.00 

86.38 

50.38     86.16 

50.75 

85.94 

51.13 

100 

0 

a 

s 

Dep. 

Lat. 

Dep. 

Lat.      Dep. 

Lat. 

Dep. 

Lat. 

1 

rt 

J 

"x 

Q 

60  Deg. 

59|  Deg. 

59*  Deg. 

59i  Deg. 

75 

Q 

64 


TRAVERSE   TABLE. 


p 

£•' 

P 

31  Deg. 

314  Deg. 

314  Deg. 

31£  Deg. 

0 
5' 

.1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

I 

1 

0.86 

0.51 

0.85 

0.52 

0.85 

0.52 

0.85 

0.53 

1 

2 

1.71 

1.03 

1.71 

1.04 

1.71 

1.04 

1.70 

1.05 

2 

3 

2.57 

1.55 

2.56 

1.56 

2.56 

1.57 

2.55 

1.58 

3 

4 

3.43 

2.06 

3.42 

2.08 

3.41 

2.09 

3.40 

2.10 

4 

5 

4.29 

2.58 

4.27 

2.59 

4.26 

2.61 

4.25 

2.63 

5 

6 

5.14 

3.09 

5.13 

3.11 

5.12 

3.13 

5.10 

3.16 

6 

7 

6.00 

3.61 

5.98 

3.63 

5.97 

3.66 

5.95 

3.68 

7 

8 

6.86 

4.12 

6.84 

4.15 

6.82 

4.18 

6.80 

4.21 

8 

9 

7.71 

4.64 

7.69 

4.67 

7.67 

4.70 

7.65 

4.74 

9 

10 

8.57 

5.15 

8.55 

5.19 

8.53 

5.22 

8.50 

5.26 

10 

11 

9.43 

5.67 

9.40 

5.71 

9.38 

5.75 

9.35 

5.79 

11 

12 

10.29 

6.18 

10.26 

6.23 

10.23 

&.  27 

10.20 

6.31 

12 

13 

11.14 

6.70 

11.11 

6.74 

11.08 

6.79 

11.05 

6.84 

13 

14 

12.00 

7.21 

11.97 

7.26 

11.94 

7.31 

11.90 

7.37 

14 

15 

12.86 

7.73 

12.82 

7.78 

12.79 

7.84 

12.76 

7.89 

15 

16 

13.71 

8.24 

13.68 

8.30 

13.64 

8.36 

13.61 

8.42 

16 

17 

14.57 

8.76 

14.53 

8.82 

14.49 

8.88 

14.46 

8.95 

17 

18 

15.43 

9.27 

15.39 

9.34 

15.35 

9.40 

15.31 

9.47 

18 

19 

16.29 

9.79 

16.24 

9.86 

16.20 

9.93 

16.16 

10.00 

19 

20 

17.14 

10.30 

17.10 

10.38 

17.05 

10.45 

17.01 

10.52 

20 

21 

18.00 

10.82 

17.95 

10.89 

17.91 

10.97 

17.86 

11.05 

21 

22 

18.86 

11.33 

18.81 

11.41 

18.76 

11.49 

18.71 

11.58 

22 

23 

19.71 

11.85 

19.66 

11.93 

19.61 

12.02 

19.56 

12.10 

23 

24 

20.57 

12.36 

20.52 

12.45 

20.46 

12.54 

20.41 

12.63 

24 

25 

21.43 

12.88 

21.37 

12.97 

21.32 

13.06 

21.26 

13.16 

25 

26 

22.29 

13.39 

22.23 

13.49 

22.17 

13.58 

22.11 

13.68 

26 

27 

23.14 

13.91 

23.08 

14.01 

23.02 

14.11 

22.96 

14.21 

27 

28 

24.00 

J4.42 

23.94 

14.53 

23.87 

14.63 

23.81 

14.73 

28 

29 

24.86 

14.94 

24.79 

15.04 

24.73 

15.15 

24.66 

15.26 

29 

30 

25.71 

15.45 

25.65 

15.56 

25.58 

15.67 

25.51 

15.79 

30 

31 

26.57 

15.97 

26.50 

16.08 

26.43 

16.20 

26.36 

16.31 

31 

32 

27.43 

16.48 

27.36 

16.60 

27.28 

16.72 

27.21 

16.84 

32 

33 

28.29 

17.00 

28.21 

17.12 

28.14 

17.24 

28.06 

17.37 

33 

34 

29.14 

17.51 

29.07 

17.64 

28.99 

17.76 

28.91 

17.89 

35 

30.00 

18.03 

29.92 

18.16 

29.84 

18.29 

29.76 

18.42 

35 

36 

30.86 

18.54 

30.78 

18.68 

30:70 

18.81 

30.61 

18.94 

36 

37 

31.72 

19.06 

31.63 

19.19 

31.55 

19.33 

31.46 

19.47 

37 

38 

32.57 

19.57 

32.49 

19.71 

32.40 

19.85 

32.31 

20.00 

38 

39 

33.43 

20.09 

33.34 

20.23 

33.25 

20.38 

33.16 

20.52 

39 

40 

34.29 

20.60 

34.20 

20.75 

34.11 

20.90 

34.01 

21.05 

40 

41 

35.14 

21.12 

35.05 

21.27 

34.96 

21.42 

34.86 

21.57 

41 

42 

36.00 

21.63 

35.91 

21.79 

35.81 

21.94 

35.71 

22.10 

42 

43 

36.86 

22.15 

36.76 

22.31 

36.66 

22.47 

36.57 

22.63 

43 

44 

37.72 

22.66 

37.62 

22.83 

37.52 

22.99 

37.42 

23.15 

44 

45 

38.57 

23.18 

38.47 

23.34 

38.37 

23.51 

38.27 

23.68 

45 

46 

39.43 

23.69 

39.33 

23.86 

39.22 

24.03 

39.12 

24.21 

46 

47 

40.29 

24.21 

40.18 

24.38 

40.07 

24.56 

39.97 

24.73 

47 

48 

41.14 

24.72 

41.04 

24.90 

40.93 

25.08 

40.82 

25.26 

48 

49 

42.00 

25.24 

41.89 

25.42 

41.78 

25.60 

41.67 

25  .  78 

49 

50 

42.86 

25.75 

42.75 

25.94 

42.63 

26.12 

42.52 

26.31 

50 

8 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D 

o 

Q 

59  Deg. 

58|  Deg. 

584  Deg. 

58i  Deg. 

1 

TRAVERSE    TABLE. 


o 

31  Deg. 

3H  Deg. 

34  Deg. 

31$  Deg. 

G 

5T 

55' 

tancc. 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

43.72 

26.27 

43.60 

26.46 

43.48 

26.65 

43.37 

26.84 

51 

52 

44.57 

26.78 

44.46 

26.98 

44.34 

27.17 

44.22 

27.36 

52 

53 

45.43 

27.30 

45.31 

27.49 

45.19 

27.69 

45.07 

27.89 

53 

54 

46.29 

27.81 

46.17 

28.01 

46.04 

28.21 

45.92 

28.42 

54 

55 

47.14 

28.33 

47.02 

28.53 

46.90 

28.74 

46.77 

28.94 

55 

56 

48.00 

28.84 

47.88 

29.05 

47.75 

29.26 

47.62 

29.47 

56 

57 

48.86 

29.36 

48.73 

29.57 

48.60 

29.78 

48.47 

29.99 

57 

58 

49.72 

29.87 

49.58 

30.09 

49.45 

30.30 

49.32 

30.52 

58 

59 

50.57 

30.39 

50.44 

30.61 

50.31 

30.83 

50.17 

31.05 

59 

60 

51.43 

30.90 

51.29 

31.13 

51.16 

31.35 

51.02 

31.57 

60 

61 

52.29 

31.42 

5'2.15 

31.65 

52.01 

31.87 

51.87 

32.10 

61 

62 

53.14 

31.93 

53.00 

32.16 

52.86 

32.39 

52.72 

32  .  63 

62 

63 

54.00 

32.45 

53.86 

32.68 

53.72 

32.92 

53.57 

33.15 

63 

64 

54.86 

32.96 

54.71 

33.20 

54.57 

33.44 

54.42 

33.68 

64 

65 

55.72 

33.48 

55.57 

33.72 

55.42 

33.96 

55.27 

34.20 

65 

66 

56.57 

33.99 

56.42 

34.24 

56.27 

34.48 

56.12 

34.73 

66 

67 

57.43 

34.51 

57.28 

34.76 

57.13 

35.01 

56.98 

35.26 

67 

68 

58.29 

35.02 

58.13 

35.28 

57.98 

35.53 

57.82 

35.78 

68 

69 

59.14 

35.54 

58.99 

35.80 

58.83 

36.05 

58.67 

36.31 

69 

70 

60.00 

36.05 

59.84 

36.31 

59.68 

36.57 

59.52 

36.83 

70 

71. 

60.86 

36.57 

60.70 

36.83 

60.54 

37.10 

60.37 

37.36 

71 

72 

61.72 

37.08 

61.55 

37.35 

61.39 

37.62 

61.23 

37.89 

72 

73 

62.57 

37.60 

62.41 

37.87 

02.24 

38.14 

62.08 

38.41 

73 

74 

63.43 

38.11 

63.26 

38.39 

63.10 

38.66 

62.93 

38.94 

74 

75 

64.29 

38.63 

64.12 

38.91 

63.95. 

39.19 

63.78 

39.47 

75 

76 

65.14 

39.14 

64.97 

39.43 

64.80 

39.71 

64.63 

39.99 

76 

77 

66.00 

39.66 

65.83 

39.95 

65.65 

40.23 

65.48 

40.52 

77 

78 

66.86 

40.17 

66.68 

40.46 

66.51 

40.75 

66.33 

41.04 

78 

79 

67.72 

40.69 

67.54 

40.98 

67.36 

41.28 

67.18 

41.57 

79 

80 

68.57 

41.20 

68.39 

41.50 

68.21 

41.80 

38.03 

42,.  10 

80 

81 

69.43 

41.72 

69.25 

42.02 

69.06 

42.32 

68.88 

42.62 

81 

82 

70.29 

42.23 

70.10 

42.54 

69.92 

42.84 

69.73 

43.15 

82 

83 

71.14 

42.75 

70.96 

43.06 

70.77 

43.37 

70.58 

43.68 

83 

84 

72.00 

43.26 

71.81 

43.58 

71.62 

43.39 

71.43 

44.20 

84 

85 

72.86 

43.78 

72.67 

44.10 

72.47 

44.41 

72.28 

44.73 

85 

86 

73.72 

44.29 

73.52 

44.61 

73.33 

44.93 

73.13 

45.25 

86 

87 

74.57 

44.81 

74.38 

45.13 

74.18 

45.46 

73.98 

45.78 

87 

88 

75.43 

45.32 

75.23 

45.65 

75.03 

45.98 

74.83 

46.31 

88 

89 

76.29 

45.84 

76.09 

46.17 

75.88 

46.50 

75.68 

46.83 

89 

90 

77.15 

46.35 

76.94 

46.69 

76.74 

47.02 

76.53 

47.36 

90 

91 

78.00 

46.87 

77.80 

47.21 

77.59 

47.55 

77.38 

47.89 

91 

92 

78.86 

47.38 

78.65 

47.73 

78.44 

48.07 

78.23 

48.41 

92 

93 

79.72 

47,90 

79.51 

48.25 

79.30 

48.59 

79.08 

48.94 

93 

94 

80.57, 

48.41 

80.36 

48,76 

80.15 

49.11 

79.93 

49.47 

94 

95 

81.43 

48.93 

81.22 

49.28 

81.00 

49.64 

80.78 

49.99 

95 

96 

82.29 

49.44 

82.07 

49.80 

81.85 

50.16 

81.63 

50.52 

96 

97 

33.15 

49.96 

82.93 

50.32 

82.71 

50.68 

82.48 

51.04 

97 

98 

84.00 

50.47 

83.78 

50.84 

83.56 

51.20 

83.33 

51.57 

98 

99 

84.86 

50,99 

84.64 

51.36 

84.41 

51.73 

84.18 

52.10 

99 

100 

85.72 

51.50 

85.49 

51.88 

85.26 

52.25 

85.04 

52.62 

100 

i* 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

^ 

cd 

to 

5 

59  Detr. 

58}  Deg. 

581  Deg. 

58i  Deg. 

5 

66 


TRAVERSE    TABLE. 


o 

32  Deg. 

32i  Deg. 

32|  Deg. 

32|  Deg. 

0 
B 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

E 

1 

0.85 

0.53 

0.85 

0.53 

0.84 

0.54 

0.84 

0.54 

T 

2 

1.70 

1.06 

1.69 

1.07 

1.69 

1.07 

1.68 

1.08 

2 

3 

2.54 

K59 

2.54 

1.60 

2.53 

1.61 

2.52 

1.62 

3 

4 

3.39 

2.12 

3.38 

2.13 

3.3~ 

2.15 

3.36 

2.16 

4 

5 

4.24 

2.65 

4.23 

2.67 

4.22 

2.69 

4.21 

2.70 

5 

6 

5.09 

3.18 

5.07 

3.20 

5.06 

3.22 

5.05 

3.25 

6 

7 

5.94 

3.71 

5.92 

3.74 

5.90 

3.76 

5.89 

3.79 

7 

8 

6.78 

4.24 

6.77 

4.27 

6.75 

4.30 

6.73 

4.33 

8 

9 

7.63 

4.77 

7.61 

4.80 

7.59 

4.84 

7.57 

4.87 

9 

10 

8.48 

5.30 

8.46 

5.34 

8.43 

5.37 

•  8.41 

5.41 

10 

11 

9.33 

5.83 

9.30 

5.87 

9.28 

5.91 

9.25 

5.95 

11 

12 

10.18 

6.36 

10.15 

6.40 

10.12 

6.45 

10.09 

6.49 

12 

13 

11.02 

6.89 

10.99 

6.94 

10.96 

6.98 

10.93 

7.03 

13 

14 

11.87 

7.42 

11.84 

7.47 

11.81 

7.52 

11.77 

7.57 

14 

15 

12.72 

7.95 

12.69 

8.00 

12.65 

8.06 

12.62 

8.11 

15 

16 

13.57 

8.48 

13.53 

8.54 

13.49 

8.60 

13.46 

8.66 

16 

17 

14.42 

9.01 

14.38 

9.07 

14.34 

9.13 

14.30 

9.20 

17 

18 

15.26 

9.54 

15.22 

9.61 

15.18 

9.67 

15.14 

9.74 

18 

19 

16.11 

10.07 

16.07 

10.14 

16.02 

10.21 

15.98 

10.28 

19 

20 

16.96 

10.60 

16.91 

10.67 

16.87 

10.75 

16.82 

10.82 

20 

21 

17.81 

11.13 

17.76 

11.21 

17.71 

11.28 

17.66 

11.36 

21 

552 

18.66 

11.66 

18.61 

11.74 

18.55 

11.82 

18.50 

11.90 

22 

23 

19.51 

12.19 

19.45 

12.27 

19.40 

12.36 

19.34 

12.44 

23 

24 

20.35 

12.72 

20.30 

12.81 

20.24 

12.90 

20.18 

12.98 

24 

25 

21.20 

13.25 

21.14 

13.34 

21.08 

13.43 

21.03 

13.52 

25 

26 

22.05 

13.78 

21.99 

13.87 

21.93 

13.97 

21.87 

14.07 

26 

27 

22.90 

14.31 

22.83 

14.41 

22.77 

14.51 

22.71 

14.61 

27 

28 

23.75 

14.84 

23.68 

14.94 

23.61 

15.04 

23.55 

15.15 

28 

29 

24.59 

15.37 

24.53 

15.47 

24.46 

15.58 

24.39 

15.69 

29 

30 

25.44 

15.90 

25.37 

16.01 

25.30 

16.12 

25.23 

16.23 

30 

31 

26.29 

16.43 

26.22 

16.54 

26.15 

16.66 

26.07 

16.77 

31 

32 

27.14 

16.96 

27.06 

17.08 

26.99 

17.19 

26.91 

17.31 

32 

33 

27.99 

17.49 

27.91 

17.61 

27.83 

17.73 

27.75 

17.85 

33 

34 

28.83 

18.02 

28  .  75 

18.14 

28.68 

18.27 

28.60 

18.39 

34 

35 

29.68 

18.55 

29.60 

18.68 

29.52 

18.81 

29.44 

18.93 

35 

36 

30.53 

19.08 

30.45 

19.21 

30.36 

19.34 

30.28 

19.48 

36 

37 

31.38 

19.61 

31.29 

19.74 

31.21 

19.88 

31.12 

20.02 

37 

38 

32.23 

20.14 

32.14 

20.28 

32.05 

20.42 

31.96 

20.56 

38 

39 

33.07 

20.67 

32.98 

20.81 

32.89 

20.95 

32.80 

21.10 

39 

40 

33.92 

21.20 

33.83 

21.34 

33.74 

21.49 

33.64 

21.64 

40 

41 

34.77 

21.73 

34.67 

21.88 

34.58 

22.03 

34.48 

22.18 

41 

42 

35.62 

22.26 

35.52 

22.41 

35.42 

22.57 

35.32 

22  .  72 

42 

43 

36.47 

22.79 

36.37 

22.95 

36.27 

23.10 

36.16 

23.26 

43 

44 

37.31 

23.32 

37.21 

23.48 

37.11 

23.64 

37.01 

23.80 

44 

45 

38.16 

23.85 

38.06 

24.01 

37.95 

24.18 

37.85 

24.34 

45 

46 

39.01 

24.38 

38.90 

24.55 

38.80 

24.72 

38.69 

24.88 

46 

47 

39.86 

24.91 

39.75 

25.08 

39.64 

25.25 

39.53 

25.43 

47 

48 

40.71 

25.44 

40.59 

25.61 

40.48 

25.79 

40.37 

25.97 

48 

49 

41.55 

25.97 

41.44 

26.15 

41.33 

26  .'33 

41.21 

26.51 

49 

50 

42.40 

26.50 

42.29 

26.68 

42.17 

26.86 

42.05 

27.05 

50 

8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

j 

§ 
§ 

£ 

' 

•5 

Q 

58  Deg. 

57|  Deg. 

57£  Deg. 

57*  Deg. 

5 

TRAVERSE    TABLE. 


67 


o 

«' 

.  i 

~51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

32  Deg. 

32*  Deg. 

32*  Deg. 

32}  Deg. 

1 

a 
o 
9 

~51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

43.25 
44.10 
4-1.95 
45.79 
46.64 
47.49 
48.34 
49.19 
50.03 
50.88 

27.03 
27.56! 
28.09  i 
28.62 
29.15 
29.68 
30.21 
30.74 
31.27 
31.80 

43.13 
43.98 
44.82 
45.67 
46.51 
47.36 
48.21 
49.00 
49.90 
50.74 

27.21, 
27.75 
28.28 
28.82 
29,35 
29.88 
30.42 
30.95 
31.48 
32.02 

43.01 
43.86 
44.70 
45.54 
46.39 
47.23 
48.07 
48.92 
49.76 
50.60 

27.40; 
27.94 
28.48: 
29.01  ! 
29.55 
30.09 
30.63 
31.16 
31.70 
32.24 

42.89 
43.73 
44.58 
45.42 
46.26 
47.10 
47.94 
48.78 
49.62 
50.46 

27.59 
28.13 
28.67 
29.21 
29.75 
30.29 
30.84 
31.38 
31.92 
32.46 

61 
62 
63 
64 
65 
66 
67 
68 
69 
70 

51.73 
52.58 
53.43 
54.28 
55.12 
55.97 
56.82 
57.67 
58.52 
59.36 

32.33] 
32.85 
33.38  i 
33.91  ! 
34.441 
34.97. 
35.50: 
36.03  | 
36.56, 
37.091 

51.59 
52.44 
53.28 
54.13 
54.97 
55.82 
56.66 
57.51 
58.36 
59.20 

32.55 
33.08 
33.62 
34.15 
34.68 
35.22 
35.75 
36.29 
36.82 
37.35 

51.45 
52.29 
53.13 
53.98 
54.82 
55.66 
56.51 
57.35 
58.19 
59.04 

32.78 
33.31 
33.85 
34.39 
34.92 
35.46 
36.00 
36.54 
37.07 
37.61 

51.30 
52.14 
52.99 
53.83 
54.67 
55.51 
56.35 
57.19 
58.03 
58.87 

33.00 
33.54 
34.08 
34.62 
35.16 
35.70 
36.25 
36.79 
37.33 
37.87 

71 
72 
73 
74 
75 
76 
77 
78 
79 
80 

81 
82 
83 
84 
85 
86 
87 
88 
89 
90 

91 
92 
93 
94 
95 
96 
97 
98 
99 
100 

60.21 
61.06 
61.91 
62.76 
63.60 
64.45 
65.30 
66.15 
67.00 
67.84 

37.62  ; 
38.15 
38.68: 
39.21  I, 
39  .  74 
40.27 
40.80 
41.33 
41.86 
42.39 

60.05 
60.89 
61.74 
62.58 
63.43 
64.28 
65.12 
65.97 
66.81 
67.66 

37.89 
38.42 
38.95 
39.49 
40:02 
40.55 
41.09 
41.62 
42.16 
42.69 

59.88 
60.72 
61.57 
62.41 
63.25 
64.10 
64.94 
65.78 
66.63 
67.47 

38.15 
38.69 
39.22 
39.76 
40.30 
40.83 
41.37 
41.91 
42.45 
42.98 

59.71 
60.55 
61.40 
62.24 
,63.08 
63.92 
'64.76 
'65.60 
'66.44 
i67.28 

38.41 
38.95 
39.49 
40.03 
40.57 
41.11 
41.65 
42.20 
42.74 
43.28 

71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 
99 
100 

68,69 
69.54 
70.39 
71.24 
72.08 
72.93 
73.78 
74.63 
75.48 
76.32 

42.92 
43.45 
43.98 
44.51 
45.04 
45.57 
46.10 
46.63 
47.16 
47.69 

68.50 
69.35 
70.20 
71.04 
71.89 
72.73 
73.58 
74.42 
75.27 
76.12 

43.22 
43.76 
44.29 
44.82 
45.36 
45.89 
46.42 
46.96 
47.49 
48.03 

68.31 
69.16 
70.00 
70.84 
71.69 
72.53 
73.38 
74.22 
75.06 
75.91 

43.521  68.12 
44.06  !  68.97 
44.60  169.81 
45.13     70.65 
45.67     71.49 
46.21     72.33 
46.75     73.17 
47.28  :  74.  01 
47.82     74.85 
48.36     75.09 

43.82 
44.36 
44.90 
45.44 
45.98 
46.52 
47.06 
47.61 
48.15 
48.69 

77.17 
78.02 
78.87 
79.72 
80.56 
81.41 
82.26 
83.11 
83.96 
84.80 

48.221 
48.75 
49.28 
49.81 
50.34 
50.87 
51.40 
51.93 
52.46 
52.99 

76.96 
77.81 
78.65 
79.50 
80.34 
81.19 
82.04 
82.88 
83.73 
84.57 

48.56 
49.09 
49.63 
50.16 
50.69 
51.23 
51.76 
52.29 
52.83 
53.36 

76.75 
77.59 
78.44 
79.28 
80.12 
80.97 
81.81 
82.65 
83.50 
84.34 

48.89     76.53 
49.43  .77.38 
49.97  178.22 
50.51     79.06 
51.04  179.90 
51.58     80.74 
52.12  181.58 
52.66  1  82.  42 
53.19  i  83.  26 
53.73     84.10 

49.23 
49  77 
50.31 
50.85 
51.39 
51.93 
52.47 
53.02 
53.56 
54.10 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.   I!  Dep. 

Lat. 

jj 

1 

58  Deg. 

57J  Deg. 

57*  Deg.            57*  Deg. 

II 

R 

68 


TRAVERSE    TABLE. 


d 

33  Deg. 

33}  Deg. 

33i  Deg. 

33|  Deg. 

C 

5' 

1' 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

1 

0.84 

0.54 

0.84 

0.55 

0.83 

0.55 

0.83 

0.56 

1 

2 

1.68 

1.09 

1.67 

1.  10 

1.67 

1.10 

1.66 

1.11 

2 

3 

2.52 

1.63 

2.51 

1.64 

2.50 

1.66 

2.49 

1.67 

3 

4 

3.35 

2.18 

3.35 

2.19 

3.34 

2.21 

3.33 

2.22 

4 

5 

4.19 

2.72 

4.18 

2.74 

4.17 

2.76 

4.16 

2.78 

5 

6 

5.03 

3.27 

5.02 

3.29 

5.00 

3.31 

4.99 

3.33 

6 

7 

5.87 

3.81 

5.85 

3.84 

5.84 

3.86 

5.82 

3.89 

7 

8 

6.71 

4.36 

6.69 

4.39 

6.67 

4.42 

6.65 

4.44 

8 

9 

7.55 

4.90 

7.53 

4.93 

7.50 

4.97 

7.48 

5.00 

9 

10 

8.39 

5.45 

8.36 

5.48 

8.34 

5.52 

8.3L 

5.56 

10 

11 

9.23 

5.99 

9.20 

6.03 

"  9.17 

6.07 

9.15 

6.11 

11 

12 

10.06 

6.54 

10.04 

6.58 

10.01 

6.62 

9.98 

6.67 

12 

13 

10.90 

7.08 

10.87 

7.13 

10.84 

7.18 

10.81 

7.22 

13 

14 

11.74 

7.62 

11.71 

7.68 

11.67 

7.73 

11.64 

7.78 

14 

15 

12.58 

8.17 

12.54 

8.22 

12.51 

8.28 

12.47 

8.33 

15 

16 

13.42 

13.38 

8.77 

13.34 

8.83 

13.30 

8.89 

16 

17 

14.26 

9!  26 

14.22 

9.32 

14.18 

9.38 

14.13 

9.44 

17 

18 

15.10 

9.80 

15.05 

9.87 

15.01 

9.93 

14.97 

10.00 

18 

19 

15.93 

10.35 

15.89 

10.42 

15.84 

10.49 

15.80 

10.56 

19 

20 

16.77 

10.89 

16.73 

10.97 

16.68 

11.04 

16.63 

11.11 

20 

21 

17.61 

11.44 

17.56 

11.51 

17.51 

11.59 

17.46 

11.6? 

21 

22 

18.45 

11.98 

18.40 

12.06 

18.35 

12.14 

18.29 

12.22 

22 

23 

19.29 

12.53 

19.23 

12.61 

19.18 

12.69 

19.12 

12.78 

23 

24 

20.13 

13.07 

20.07 

13.16 

20.01 

13.25 

19.96 

13.33 

24 

25 

20.97 

13.62 

20.91 

13.71 

20.85 

13.80 

20.79 

13.89 

25 

26 

21.81 

14.16 

21.74 

14.26 

21:68 

14.35 

2\.  62 

14.44 

26 

27 

22.64 

14.71 

22.58 

14.80 

22.51 

14.90 

22.45 

15.00 

27 

28 

23.48 

15.25 

23.42 

15.35 

23.35 

15.45 

23.28 

15.56 

28 

29 

24.32 

15.79 

24.25 

15.90 

24.18 

16.01 

24.11 

16.11 

29 

30 

25.16 

16.34 

25.09 

16.45 

25.02 

16.56 

24.94 

16.67 

30 

31 

26.00 

16.88 

25.92 

17,00 

25.85 

17.11 

25.78 

17.22 

31 

32 

26.84 

17.43 

26.76 

17.55 

26.68 

17.66 

26.61 

17.78 

32 

33 

27.68 

17.97 

27.60 

18.09 

27.52 

18.21 

27.44 

18.33 

33 

34 

28.51 

18.52 

28.43 

18.64 

28.35 

18.77 

28.  2# 

13.89 

34 

35 

29.35 

19.06 

29.27 

19.19 

29.19 

19.32 

29.10 

19.44 

35 

36 

30.19 

19.61 

30.11 

19.74 

30.02 

19.87 

29.93 

20.00 

36 

37 

31.03 

20.15 

30.94 

20.29 

30.85 

20.42 

30.76 

20.56 

37 

38 

31.87 

2,0.70 

31.78 

20.84 

>31.69 

20.97 

31.60 

21.11 

38 

39 

32  71 

21.24 

32.62 

21.38 

32.52 

21.53 

32.43 

21.67 

39 

40 

33.55 

21.79 

33.45 

21.93 

33.36 

22.08 

33.26 

22.22 

40 

41 

34.39 

22.33 

34.29 

22.48 

34'.  19 

22.63 

34.09 

22.78 

41 

42 

35.22 

22.87 

35.12 

23.03 

35.02 

23.18 

34.92 

23.33 

42 

43 

36.06 

23.42 

35.96 

23.58 

35.86 

23.73 

35.75 

23.89 

43 

44 

36.90 

23.96 

36.80 

24.12 

36.69 

24.29 

36.58 

24.45 

44 

45 

37.74 

24.51 

37.63 

24.67 

37.52 

24.84 

37.42 

25.00 

45 

46 

38.58 

25.05 

38.47 

25.22 

38.36 

25.39 

38.25 

25.56 

46 

47 

39.42 

25.60 

39.31 

25.77 

39.19 

25.94 

39.08 

26.11 

47 

48 

40.26 

26.14 

40.14 

26.32 

40.03 

26.49 

39.91 

26.67 

48 

49 

41.09 

26.69 

40.98 

26.87 

40.86 

27.04 

40.74 

27.22 

49 

50 

41.93 

27.23 

41.81 

27.41 

41.69 

27.60 

41.57 

27.78 

50 

D 

O 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

d 

o 

d 

d 

ci 

3 

"w 

b 

57  Deg. 

56J  Deg. 

56i  Deg. 

56J  Deg. 

3 

TRAVERSE    TABLE. 


69 


g 

33  Deg. 

33i  Deg. 

33£  Deg. 

33|  Deg. 

a 

nance. 

stance. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

Lat. 

Dep. 

51 

42.77 

27.78 

42.65 

27.96 

42.53 

28.1J> 

42.40 

28.33 

51 

52 

43.61 

28.32 

43.49 

28.51 

43.36 

28.70 

43.24 

28.89 

52 

53 

44.45 

28.87 

44.32 

29.06 

44.20 

29.25 

44.07 

29.45 

53 

54 

45.29 

29.41 

45.16 

29.61 

45.03 

29.80 

44.90 

30.00 

54 

55 

46.13 

29.96 

46.00 

30.16 

45.86 

30.36 

45.73 

30.56 

55 

56 

46.97 

30.50 

46.83 

30.70 

46.70 

30.91 

46.56 

31.11 

56 

57 

47.80 

31.04 

47.67 

31.25 

47.53 

31.46 

47.39 

31.67 

57 

58 

48.64 

31.59 

48.50 

31.80 

48.37 

32.01 

48.23 

32.22 

58 

59 

49.48 

32.13 

49.34 

32.35 

49.20 

32.56 

49.06 

32.78 

59 

60 

50.32 

32.68 

50.18 

32.90 

50.03 

33.12 

49.89 

33.33 

60 

61 

51.16 

33.22 

51.01 

33.45 

50.87 

33.67 

50.72 

33.89 

61 

62 

52.00 

33.77 

51.85 

33.99 

51.70 

34.22 

51.55 

34.45 

62 

63 

52.84 

34.31 

52.69 

34.54 

52.53 

34.77 

52.38 

35.00 

63 

64 

53.67 

34.86 

53.52 

35.09 

53.37 

35.32 

53.21 

35.56 

64 

65 

54.51 

35.40 

54.36 

35.64 

54.20 

35.88 

54.05 

36.11 

65 

66 

55.35 

35.95 

55.19 

36.19 

55.04 

36.43 

54.88 

36.67 

66 

67 

56.19 

36.49 

56.03 

36.74 

55.87 

36.98 

55.71 

37.22 

67 

68 

57.03 

37.04 

56.87 

37.28 

56.70 

37.53 

56.54 

37.78 

68 

69 

57.87 

37.58 

57.70 

37.83 

57.54 

38.08 

57.37 

3.8.33 

69 

70 

58.71 

38.12 

58.54 

38.38 

58.37 

38.64 

58.20 

38.89 

70 

71 

59.55 

38.67 

59.38 

38.93 

59.21 

39.19 

59.03 

39.45 

71 

72 

60.38 

39.21 

60.21 

39.48 

60.04 

39.74 

59.87 

40.00 

72 

73 

61.22 

39.76 

61.05 

40.03 

60.87 

40.29 

60.70 

40.56 

73 

74 

62.06 

40.30 

61.89 

40.57 

61.71 

40.84 

61.53 

41.11 

74 

75 

62.90 

40.85 

62.72 

41.12 

62.54 

41.40 

62.36 

41.67 

75 

76 

63.74 

41.39 

63.56 

41.67 

63.38 

41.95 

63.19 

42.22 

76 

77 

64.58 

41.94 

64.39 

42.22 

64.21 

42.50 

64.02 

42.78 

77 

78 

65.42 

42.48 

65.23 

42.77 

65.04 

43.05 

64.85 

43.33 

78 

79 

66.25 

43.03 

66.07 

43.32 

65.88 

43.60 

65.69 

43.89 

79 

80 

67.09 

43.57 

66.90 

,43.86 

66.71 

44.15 

66.52 

44.45 

80 

81 

67.93 

44.12 

67.74 

44.41 

67.54 

44.71 

67.35 

45.00 

81 

82 

68.77 

44.66 

68.58 

44.96 

68.38 

45.26 

68.18 

45.56 

82 

83 

69.61 

45.20 

69.41 

45.51 

69.21 

45.81 

69.01 

46.11 

83 

84 

70.45 

45.75 

70.25 

46.06 

70.05 

46.36 

69.84 

46.67 

84 

85 

71.29 

46.29 

71.08 

46.60 

70.88 

46.91 

70.67 

47.22 

85 

86 

72.13 

46.84 

71.92 

47.15 

71.71 

47.47 

71.51 

47.78 

86 

87 

72.96 

47.38 

72.76 

47.70. 

72.55 

48.02 

72.34 

48.33 

87 

88 

73.80 

47.93 

73.59 

48/.25< 

73.38 

48.57 

73.17 

48.89 

88 

89 

74.64 

48.47 

74.43 

48.80 

74.22 

49.12 

74.00 

49.45 

89 

90 

75.48 

49.02 

75.27 

49.35 

75.05 

49.67 

74.83 

50.00 

90 

91 

76.32 

49.56 

76.10 

49.89 

75.88 

50.23 

75.66 

50.56 

91 

92 

77.16 

50.11 

76.94 

50.44 

76.72 

50.78 

76.50 

51.11 

92 

93 

78.00 

50.65 

77.77 

50.99 

77.55 

51.33 

77.33 

51.67 

93 

94 

78.83 

51.20 

78.61 

51.54 

78.39 

51.88 

78.16 

52.22 

94 

95 

79.67 

51.74 

79.45 

52.09 

79.22 

52.43 

78.99 

52.78 

95 

96 

80.51 

52.29 

80.28 

52.64 

80.05 

52.99 

79.82 

53.33 

96 

97 

81.35 

52.83 

81.12 

53.18 

80.89 

53.54 

80.65 

53.89 

97 

98 

82.19 

53  .  37 

81.96 

53.73 

81.72 

54.09 

81.48 

54.45 

98 

99 

83.03 

53.92 

82.79 

54.28 

82.55 

54.64 

82.32 

55.00 

99 

100 

83.87 

54.46 

83.63 

54.83 

83.39 

55.19 

83.15 

55.56 

100 

§ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

.2 

a 

57  Deg. 

56|  Deg. 

56£  Deg. 

56i  Deg. 

• 

70 


TBAVERSE    TABLE. 


0 

34  Deg. 

344  Deg. 

34|  Deg. 

34|  Deg. 

O 

g. 

a* 
P 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

1 

0.83 

0.56 

0.83 

0.56 

0.82 

0.57 

0.82 

0.57 

i 

2 

1.66 

1.12 

1.65 

1.13 

1.65 

1.13 

1.64 

1.14 

2 

3 

2.49 

1.68 

2.48 

1.69 

2.47 

1.70 

2.46 

1.71 

3 

4 

3.32 

2.24 

3.31 

2.25 

3.30 

2.27 

3.29 

2.28 

4 

5 

4.15 

2.80 

4.13 

2.81 

4.12 

2.83 

4.11 

2.85 

5 

6 

4.97 

3.36 

4.96 

3.38 

4.94 

3.40 

4.93 

3.42 

6 

7 

5.80 

3.91 

5.79 

3.94 

5.77 

3.96 

5.75 

3.99 

7 

8 

6.63 

4.47 

6.61 

4.50 

6.59 

4.53 

6.57 

4.56 

8 

9 

7.46 

5.03 

7.44 

5.07 

7.42 

5.10 

7.39 

5.13 

9 

10 

8.29 

5.59 

8.27 

5.63 

8.24 

5.66 

8.22 

5.70 

10 

11 

9.12 

6.15 

9.09 

6.19 

9.07 

6.23 

9.04 

6.27 

11 

12 

9.95 

6.71 

9.92 

6.75 

9.89 

6.80 

9.86 

6.84 

12 

13 

10.78 

7.27 

10.75 

7.32 

10.71 

7.36 

10.68 

7.41 

13 

14 

11.61 

7.83 

11.57 

7.88 

11.54 

7.93 

11.50 

7.98 

14 

15 

12.44 

8.39 

12.40 

8.44 

12.36 

8.50 

12.32 

8  .  55 

15 

16 

13.26 

8.95 

13.23 

9.00 

13.19 

9.06 

13.15 

9.12 

16 

17 

14.09 

9.51 

14.05 

9.57 

14.01 

9.63 

13.97 

9.69 

17 

18 

14.92 

10.07 

14.88 

10.13 

14.83 

10.20 

14.79 

10.26 

18 

19 

15.75 

10.62 

15.71 

10.69 

15.66 

10.76 

15.61 

10.83 

19 

20 

16.58 

11.18 

16.53 

11.26 

16.48 

11.33 

16.43 

11.40 

20 

21 

17.41 

11.74 

17.36 

11.82 

17.31 

11.89 

17.25 

11.97 

21 

22 

18.24 

12.30 

18.18 

12.38 

18.13 

12.46 

18.08 

12.54 

22 

23 

19.07 

12.86 

19.01 

12.94 

18.95 

13.03 

18.90 

13.11 

23 

24 

19.90 

13.42 

19.84 

13.51 

19.78 

13.59 

19.72 

13.68 

24 

25 

20.73 

13.98 

20.66 

14.07 

20.60 

14.16 

20.54 

14.25 

25 

26 

21.55 

14.54 

21.49 

14.63 

21.43 

14.73 

21.36 

14.82 

26 

27 

22.38- 

15.10 

22.32 

15.20 

22.25 

15.29 

22.18 

15.39 

27 

28 

23.21 

15  66 

23.14 

15.76 

23.08 

15.86 

23.01 

15.96 

28 

29 

24.04 

16.22 

23.97 

16.32 

23.90 

16.43 

23.83 

16.53 

29 

30 

24.87 

16.78 

24.80 

16.88 

24.72 

16.99 

24.65 

17.10 

30 

31 

25.70 

17.33 

25  .  62 

17.45 

25  .  55 

17.56 

25.47 

17.67 

31 

32 

26.53 

17.89 

26.45 

18.01 

26  .  37 

18.12 

26.29 

18.24 

32 

33 

27.36 

18.45 

27.28 

18.57 

27.20 

18.69 

27.11 

18.81 

33 

34 

28.19 

19.01 

28.10 

19.14 

28.02 

19.26 

27.94 

19.38 

34 

35 

29.02 

19.57 

28.93 

19.70 

28.84 

19.82 

28.76 

19.95 

33 

36 

29.85' 

20.13 

29.76 

20.26 

29.67 

20.39 

29.58 

20.52 

36 

37 

30  .  67 

20.69 

30.58 

20.82 

^0.49 

20.96 

30.40 

21.09 

37 

38 

31.50 

21.25 

31.41 

21.39 

il.88 

21.52 

31.22 

21.66 

38 

39 

32.33 

21.81 

32.24 

21.95 

32.14 

22.09 

32.04 

22.23 

39 

40 

33.16 

22.37 

33.06 

22.51 

32.97 

22.66 

32.87 

22  .  80 

40 

41 

33.99 

22.93 

33.89 

23.07 

33.79 

23.22 

33.69 

23.37 

41 

42 

34.82 

23.49 

34.72 

23.64 

34.61 

23.79 

34.51 

23.94 

42 

43 

35.65 

24.05 

35.54 

24.20 

35.44 

24.36 

35.33 

24.51 

43 

44 

36.48 

24.60 

36.37 

24.76 

36.26 

24.92 

36.15 

25  .  08 

44 

45 

37.31 

35.16 

37.20 

25.33 

37.09 

25.49 

36.97 

25.65 

45 

46 

38.14 

25.72 

38.02 

25.89 

37.91 

26.05 

37.80 

26.22 

46 

47 

38  .  96 

26.28 

38.85 

26.45 

38.73 

26.62 

38.62 

26.79 

47 

48 

39.79 

26.84 

39.68 

27.01 

39.56 

27.19 

39.44 

27.36 

48 

49 

40.62 

27.40 

40.50 

27.58 

40.38 

27.75 

40.26 

27.93 

49 

50 

41  .45 

27.96 

41.33 

28.14 

41.21 

28.32 

41.08 

28  .  50 

50 

1 

De| 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.. 

i 

.3 

Q 

66  Deg. 

55|  Deg. 

55£Deg. 

554  Deg. 

cd 

3 

TRAVERSE    TABLE. 


71 


2 

34Deg. 

34*  Deg. 

34A  Deg. 

34|  Deg. 

b 

5' 

i 

I' 

a 
n 
? 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

42.28 

28.52; 

42.16 

28.70 

42.03 

28.89 

41.90 

29.07 

51 

52 

43.11 

29.08 

42.98 

29.27 

42.85 

29.45 

42.73 

29.64 

52 

53 

43.94 

29.64 

43.81 

29.83 

43.68 

30.02 

43.55 

30.21 

53 

54 

44.77 

30.20 

44.64 

30.39 

44.50 

30.59 

44.37 

30.78 

54 

55 

45.60 

30.76 

45.46 

30.95 

45.33 

31.15 

45.19 

31.35 

55 

56 

46.43 

31.31 

46.29 

31.52 

46.15 

31.72 

46.01 

31.92 

56 

57 

47.26 

31.87. 

47.12 

32.08 

46.98 

32.29 

46.83 

32.49 

57 

58 

48.08 

32.43 

47.94 

32.64 

47.80 

32.85 

47.66 

33.06 

58 

59 

48.91 

32.99 

48.77 

33.21 

48.62 

33.42 

48.48 

33.63 

59 

60 

49.74 

33.55 

49.60 

33.77 

49.45 

33.98 

49.30 

34.20 

60 

61 

50.57 

34.11 

50.42 

34.33 

50.27 

34.55 

50.12 

34.77 

61 

62 

51.40 

34.67 

51.25 

34.89 

51.10 

35.12 

50.94 

35.34 

62 

63 

52.23 

35.23 

52.08 

35.46 

51.92 

35.68 

51.76 

35.91 

63 

64 

53.06 

35.79 

52.90 

36.02 

52  .  74 

36.25 

52.59 

36.48 

64 

65 

53.89 

36.35 

53.73 

36.58 

53.57 

36.82 

53.41 

37.05 

65 

66 

54.72 

36.91 

54.55 

37.15 

54.39 

37.38 

54.23 

37.62 

66 

67 

55.55 

37.46 

55.38 

37.71 

55.22 

37.95 

55.05 

38.19 

67 

68 

56.37 

38.03 

56.21 

38.27 

56.04 

38.52 

55.87 

38.76 

68 

69 

57.20 

38.58 

57.03 

38.83 

56.86 

39.08 

56.69 

39.33 

69 

70  158.03 

39.14 

57.86 

39.40 

57.69 

39.65 

57.52 

39.90 

70 

71 

58.86 

39.70 

58.69 

39.96 

58.51 

40.21 

58.34 

40.47 

71 

72 

59.69 

40.26 

59.51 

40.52 

59.34 

40.78 

59.16 

41.04 

72 

73 

60.52 

40.82 

60.34 

41.08 

60.16 

41.35 

59.98 

41.61 

73 

74 

61.35 

41.38 

61.17 

41.65 

60.99 

41.91 

60.80 

42.18 

74 

75 

62.18 

41.94 

61.99 

42.21 

61.81 

42.48 

61.62 

42.75 

75 

76 

63.01 

42.50 

63.82 

42.77 

62.63 

43.05 

62.45 

43.32 

76 

77 

63.84 

43.06 

63.65 

43.34 

63.46 

43.61 

63.27 

43.89 

77 

78 

64.66 

43  .  62 

64.47 

43.90 

64.28 

44.18 

64.09 

44.46 

78 

79 

65.49 

44.18 

65.30 

44.46 

65  .  1  1 

44.75 

64.91 

45.03 

79 

80 

66.32 

44.74 

66.13 

45.02 

65.93 

45.31 

65.73 

45.60 

80 

81 

67.15 

45.29 

66.95 

45.59 

66.  ?o 

45.88 

66.55 

46.17 

81 

82 

67.98 

45.85 

67.78 

46.15 

67.58 

46.45 

67.37 

46.74 

82 

83 

68.81 

46.41 

68.61 

46.71 

68.40 

47.01 

68.20 

47.31 

83 

84 

69.64 

46.97 

69.43 

47.28 

69,23 

47.58 

69.02 

47.88 

84 

85 

70.47 

47.53 

70.26 

47.84 

70.05 

48.14 

69.84 

48.45 

85 

86 

71.30 

48.09 

71.09 

48.40 

70.87 

48.71 

70.66 

49.02 

86 

87 

72.13 

48.65 

71.91 

48.96 

71.70 

49.28 

71.48 

49.59 

87 

88 

72.96 

49.21 

72.74 

49.53 

72.52 

49.84 

72.30 

50.16 

88 

89 

73.78 

49.77 

73.57 

50.09 

73.35 

50.41 

73.13 

50.73 

89 

90 

74.61 

50.33 

74.39 

50.65 

74.17 

50.98 

73.95 

51.30 

90 

91 

75.44 

50.89 

75.22 

51.22 

75.00 

51.54 

74.77 

51.87 

91 

92 

76.27 

51.45 

76.05 

51.78 

75.82 

52.11 

75.59 

52.44 

92 

93 

77.10 

52.00 

76.87 

52.34 

76.64 

52.68 

76.41 

53.01 

93 

94 

77.93 

52.56 

77.70 

52.90 

77.47 

53.24 

77.23 

53.58 

94 

95 

78.76 

53.12 

78.53 

53.47 

78.29 

53.81 

78.06 

54.15 

95 

96 

79.59 

53.68 

79.35 

54.03 

79.12 

54.37 

78.88 

54.72 

96 

97 

80.42 

54.24 

80.18 

54.59 

79.94 

54.94 

79.70 

55.29 

97 

98 

81.25 

54.80 

81.01 

55.15 

80.76 

55.51 

80.52 

55.86 

98 

99 

82.07 

55.36 

81.83 

55.72 

81.59 

56.07 

81.34 

56.43 

99 

100 

82.90 

55.92 

82.66 

56.28 

82.41 

56.64 

82.16 

57.00 

100 

\ 

Dep. 

Lat/ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 
o 
c 

a 

56  Deg. 

55|  Deg. 

551  Deg. 

55i  Deg. 

Cd 

'jo 

3 

72 


TRAVERSE   TABllE. 


s 

35  Deg. 

35*  Deg. 

35i  Deg. 

35|  Deg. 

2 

1 

s» 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

p 

1 

0.82 

0.57 

0.82 

0.58 

0.81 

0.58 

0.81 

0.58 

f 

2 

1.64 

1.15 

1.63 

1.15 

1.63 

1.16 

1.62 

1.17 

2 

3 

2.46 

1.72 

2.45 

1.73 

2.44 

1.74 

2.43 

1.75 

3 

4 

3.28 

2.29 

3.27 

2.31 

3.26 

2.32 

3.25 

2.34 

4 

5 

4.10 

2.87 

4.08 

2.89 

4.07 

2.90 

4.06 

2.92 

5 

6 

4.91 

"3.44 

4.90 

3.46 

4.88 

3.48 

4.87 

3.51 

6 

7 

5.73 

4.01 

5.72 

4.04 

5.70 

4.06 

5.68 

4.09 

7 

8 

6.55 

4.59 

6.53 

4.62 

6.51 

4.65 

6.49 

4.67 

8 

9 

7.37 

5.16 

7.35 

5.19 

7.33 

5.23 

7.30 

5.26 

9 

10 

8.19 

5.74 

8.17 

5.77 

8.14 

5.81 

8.12 

5.84 

10 

"  11 

9.01 

0.31 

8.98 

6.35 

8.96 

6.39 

8.93 

6.43 

11 

12 

9.83 

6.88 

9.80 

6.93 

9.77 

6.97 

9.74 

7.01 

12 

13 

10.65 

7.46 

10.62 

7.50 

10.58 

7.55 

10.55 

7.60 

13 

14 

11.47 

8.03 

11.43 

8.08 

11.40 

8.13 

11.36 

8.18 

14 

15 

12.29 

8.60 

12.25 

8.66 

12.21 

8.71 

12.17 

8.76 

15 

16 

13.11 

9.18 

13.07 

9.23 

13.03 

9.29 

12.99 

9.35 

16 

17 

13.93 

9.75 

13.88 

9.81 

13.84 

9.87 

13.80 

9.93 

17 

18 

14.74 

10.32 

14.70 

10.39 

14.65 

10.45 

14.61 

10.52 

18 

19 

15.56 

10.90 

15.52 

10.97 

15.47 

11.03 

15.42 

11.10 

19 

20 

16.38 

11.47 

16.33 

11.54 

16.28 

11.61 

16.23 

11.68 

20 

21 

17.20 

12.05 

17.15 

12.12 

17.10 

12.19 

17.04 

12.27 

21 

22 

18.02 

12.62 

17.97 

12.70 

17.91 

12.78 

17.85 

12.85 

22 

23 

18.84 

13.19 

18.78 

13.27 

18.72 

13.36 

18.67 

13.44 

23 

24 

19.66 

13.77 

19.60 

13.85 

19.54 

13.94 

19.48 

14.02 

24 

25 

20.48 

14.34 

20.42 

14.43 

20.35 

14.52 

20.29 

14.61 

25 

26 

21.30 

14.91 

21.23 

15.01 

21.17 

15.10 

21.10 

15.19 

26 

27 

22.12 

15.49 

22.05 

15.58 

21.98 

15.68 

21.91 

15.77 

27 

28 

22.94 

16.06 

22.87 

16.16 

22.80 

16.26 

22.72 

16.36 

28 

29 

23.76 

16.63 

23.68 

16.74 

23.61 

16.84 

23.54 

16.94 

29 

30 

24.57 

17.21 

24.50 

17.31 

24.42 

17.42 

24.35 

17.53 

30 

31 

25.39 

17.78 

25.32 

17.89 

25.24 

18.00 

25.16 

18.11 

31 

32 

26.21 

18.35 

26.13 

18.47 

26.05 

18.58 

25.97 

18.70 

32 

33 

27.03 

18.93 

26.95 

19.05 

26.87 

19.16 

26.78 

19.28 

33 

34 

27.85 

19.50 

27.77 

1  9  .  62 

27.68 

19.74 

27.59 

19.86 

34 

35 

28.67 

20.08 

28.58 

20.20 

28.49 

20.32 

28.41 

20.45 

35 

36 

29.49 

20.65 

29.40 

20.78 

29.31 

20.91 

29.22 

21.03 

36 

37 

30.31 

21.22 

30.22 

21.35 

30.12 

21.49 

30.03 

21.62 

37 

38 

31.13 

21.80 

31.03 

21.93 

30.94 

22.07 

30.84 

22.20 

38 

39 

31.95 

22.37 

31.85 

22.51 

31.75 

22.65 

31.65 

22.79 

39 

40 

32.77 

22.94 

32.67 

23.09 

32.56 

23.23 

32.46 

23.37 

40 

41 

33.59 

23.52 

33.48 

23.66 

33.  3S 

23.81 

33.27 

23.95 

41 

42 

34.40 

24.09 

34.30 

24.24 

34.19 

24.39 

34.09 

24.54 

42 

43 

35.22 

24.66 

35.12 

24.82, 

35.01 

21.97 

34.90 

25.12 

43 

44 

36.04 

25  .  24 

35.93 

25.39 

35.82 

25.55 

35.7? 

25.71 

44  ' 

45 

36.86 

25.81 

36.75 

25.97 

36.64 

26.13 

36.52 

26.29 

45 

46 

37.68 

26.38 

37.57 

26.55 

37.45 

26.71 

37.33 

26.88 

46 

47 

38.50 

26.96 

38.38 

27.13 

38.26 

27.29 

38.14 

27.46 

47 

48 

39.32 

27.53 

39.20 

27.70 

39.08 

27.87 

38.96 

28.04 

48 

49 

40.14 

28.11 

40.02 

28.28 

39.89 

28.45 

39.77 

28.63 

49 

50 

40.96 

28.68 

40.83 

28.86 

40.71 

29.04 

40.58 

29.21 

50 

6 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

•Dep. 

Lat. 

d 

o 

SM 

S 

B 

3 

55  Deg. 

54|  Deg. 

54^  Deg. 

54i  Deg. 

*2 

3 

THAVERSE   TABLE. 


o 

35Deg. 

35i  Deg. 

35i  Deg. 

35|  Deg. 

C 

ft 

p 

5° 

8 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

41.78 

29.25 

41.65 

29.43 

41.52 

29.62 

41.39 

29.80 

51 

52 

42.60 

29.83 

42.47 

30.01 

42.33 

30.20 

42.20 

30.38 

52 

53 

43.42 

30.40 

43.28 

30.59 

43.15 

30.78 

43.01 

30.97 

53 

54 

44.23 

30.97 

44.10 

31.17 

43.96 

31.36 

43.82 

31.55 

54 

55 

45.05 

31.55 

44.92 

31.74 

44.78 

31.94 

44.64 

32.13 

55 

56 

45.87 

32.12 

45.73 

32.32 

45.59 

32.52 

45.45 

32.72 

56 

57 

46.69 

32.69 

46.55 

32.90 

46.40 

33.10 

46.26 

33.30 

57 

58 

47.51 

33.27 

47.37 

33.47 

47.22 

33.68 

47.07 

33.89 

58 

59 

48.33 

33.84 

48.18 

34.05 

48.03 

34.26 

47.88 

34.47 

59 

60 

49.15 

34.41 

49.00 

34.63 

48.85 

34.84 

48.69 

35.05 

60 

61 

49.97 

34.99 

49.82 

35.21 

49.66 

35.42 

49.51 

35.64" 

61 

62 

50.79 

35.56 

50.63 

35.78 

50.48 

36.00 

50.32 

36.22 

62 

63 

51.61 

36.14 

51.45 

36.36 

51.^29 

36.58 

51.13 

36.81 

63 

64 

52.43 

36.71 

52.27 

36.94 

52.10 

37.16 

51.94 

37.39 

64 

65 

53.24 

37.28 

53.08 

37.51 

52.92 

37.75 

52.75 

37.98 

65 

66 

54.06 

37.86 

53.90 

38.09 

53.73 

38.33 

53.56 

38.56 

66 

67 

54.88 

38.43 

54.71 

38.67 

54.55 

38.91 

54.38 

39.14 

67 

68 

55.70 

39.00 

55.53 

39.  .25 

55.36 

39.49 

55.19 

39.73 

68 

69 

56.52 

39.58 

56.35 

39.82 

56.17 

40.07 

56.00 

40.31 

69 

70 

57.34 

40.15 

57.16 

40  40 

56.99 

40.65 

56.81 

40.90 

70 

71 

58.16 

40.72 

57.98 

40.98 

57.80 

41.23 

57.62 

41.48 

71 

72 

58.98 

41.30 

58.80 

41.55 

58  .  62 

41.81 

58.43 

42.07 

72 

73 

59.80 

41.87 

59.61 

42.13 

59.43 

42.39 

59.24 

42.65 

73 

74 

60.  62 

42.44 

60.43 

42.71 

60.24 

42.97 

60.06 

43.23 

74 

75 

6T.44 

43.02 

61.25 

43.29 

61.06 

43.55 

60.87 

43.82 

75 

76 

62.26 

43.59 

62.06 

43.86 

61.87 

44.13 

61.68 

44.40  •  76 

77 

63.07 

44.17 

62.88 

44.44 

62.69 

44.71 

62.49 

44.99 

77 

78 

63.89 

44.74 

63.70 

45.02 

63.50 

45.29 

63.30 

45.57 

78 

79 

64.71 

45.31 

64.51 

45.59 

64.32 

45.88 

64.11 

46.16 

79 

80 

65.53 

45.89 

65.33 

46.17 

65.13 

46.46 

64.93 

46.74 

8Q 

81 

66.35 

46.46 

66.15 

46.75 

65.94 

47.04 

65.74 

47.32 

81 

82 

67.17 

47.03 

66.96 

47.33 

66.76 

47.62 

66.55 

47.91 

82 

83 

67.99 

47.61 

67.78 

47.90 

67.57 

48.20 

67.36 

48.49 

83 

84 

68.81 

48.18 

68.60 

48.48! 

68.39 

48.78 

68.17 

49.08 

84 

85 

69.63 

48.75 

69.41 

49.  06';  69.  20 

49.  3& 

68.98 

49.66 

85 

86 

70.45 

49.33 

70.23 

49.63  i  70.  01 

49.94 

69.80 

50.25 

86 

87 

71.27 

49.90 

71.05 

50.21  I  70.83 

50.52 

70.61 

50.83 

87 

88 

72.09 

50  47 

71.86 

50.79 

71.64 

51.10 

71.42 

51.41 

88 

89 

72.90 

51.05 

72.68 

51.37 

72.46 

51.68 

72.23 

52.00 

89 

90 

73.72 

51.62 

73.50 

51.94 

73.27 

52.26  ! 

73.04 

52.58 

90 

91 

74.54 

52.20 

74.31 

52.52 

74.08 

52.84 

73.85 

53.17 

91 

92 

75.36 

52.77 

75.13 

53.10 

74.90 

53.42 

74.66 

53.75 

92 

93 

76.18 

53.34 

75.95 

53.67 

75.71 

54.01 

75.48 

54.34 

93 

94 

77.00 

53.92 

76.76 

54.25 

76.53 

54.59 

76.29 

54.92 

94 

95 

77.82 

54.49 

77.58 

54.83 

77.34 

55.17 

77.10 

55.50 

95 

96 

78.64 

55.06 

78.40 

55.41 

78.16 

55.75 

77.91 

56.09 

96 

97 

79.46 

55.64 

79.21 

55.98 

78.97 

56.33 

78.72 

56.67 

97 

98 

80.28 

56.21 

80.03 

56.56 

79.78 

56.91 

79.53 

57.26 

98 

99 

81.10 

56.78 

80.85 

57.14 

80.60 

57.49 

80.35 

57.84 

99 

100 

81.92 

57.36 

81.66 

57.71 

81.41 

58.07 

81.16 

58.42 

100 

• 

Q 

e 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

OJ 

CJ 

c 

3 

d 

s 

55  Deg. 

54|  Deg. 

54iDeg. 

544  Deg. 

5 

74 


TRAVERSE    TABLE. 


G  1       36  Deg. 

cc*  1 

36i  Deg. 

36|  Deg. 

36|  Deg. 

5' 

1 

I? 

I  1    L/at. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

P 

1 

0.81 

0.59 

0.81 

0.59 

0.80 

0.59 

0.80 

0.60 

1 

2 

1.62 

1.18 

1.61 

1.18 

1.61 

1.19 

1.60 

1.20 

2 

3 

2.43 

1.76 

2.42 

1.77 

'  2.41 

1.78 

2.40 

1.79 

3 

4 

3.24 

2.35 

3.23 

2.37 

3.22 

2.38 

3.20 

2.39 

4 

5 

4.05 

2.94 

4.03 

2.96 

4.02 

2.97 

4.01 

2.99 

5 

6 

4.85 

3.53 

4.84 

3.55 

4.82 

3.57 

4.81 

3.59 

6 

7 

5.66 

4.11 

5.65 

4.14 

5.63 

4.16 

5.61 

4.19 

7 

8 

6.47 

4.70 

6.45 

4.73 

6.43 

4.76 

6.41 

4.79 

8 

9 

7.28 

5.29 

7.26 

5.32 

7.23 

5.35 

7.21 

5.3S 

9 

10 

8.09 

5.88 

8.06 

5.91 

8.04 

5.95 

8.01 

5.98 

10 

1] 

8.90 

6.47 

8.87 

6.50 

8.84 

6.54 

8.81 

6.58 

11 

12 

9.71 

7.05 

9.68 

7.10 

9.65 

7.14 

9.61 

7.18 

12 

13 

10.52 

7.64 

10.48 

7.69 

10.45 

7.73 

-10.42 

7.78 

13 

14 

11.33 

8.23 

11.29 

8.28 

11.25 

8.33 

11.22 

8.38 

14 

15 

12.14 

8.82* 

12.10 

8.87 

12.06 

8.92 

12.02 

8.97 

15 

16 

12.94 

9.40 

12.90 

9.46 

12.86 

9.52 

12.82 

9.57 

16 

17 

13.75 

9.99 

13.71 

10.05 

13.67 

10.11 

13.62 

10.17 

17 

18 

14.56 

10.58 

14.52 

10.64 

14.47 

10.71 

14.42 

10.77 

18 

19 

15.37 

11.17 

15.32 

11.23 

15.27 

11.30 

15.22 

11.37 

19 

20. 

16.18 

11.76 

16.13 

11.83 

16.08 

11.90 

16.03 

11.97 

20 

21 

16.99 

12.34 

16.94 

12.42 

16.88 

12.49 

16.83 

12.56 

21 

22 

17.80 

12.93 

17.74 

13.01 

17.68 

13.09 

17.63 

13.16 

22 

23 

18.61 

13.52 

18.55 

13.60 

18.49 

13.68 

18.43 

13.76 

23 

24 

19.42 

14.11 

19.35 

14.19 

19.29 

14.  28 

.19.23 

14.36 

24 

25 

20.23 

14.69 

20.16 

14.78 

20.10 

.14.87 

20.03 

14.96 

25 

26 

21.03 

15.28 

20.97 

15.37 

20.90 

15.47 

20.83 

15.56 

26 

27 

21.84 

15.87 

21.77 

15.97 

21.70 

16.06 

21.63 

16.15 

27 

28 

22  .  65 

16.46 

22.58 

16.56 

22.51 

16.65 

22.44 

16.75 

28 

29 

23.46 

17.05 

23.39 

17.15 

23.31 

17.25 

23.24 

17.35 

29 

•30 

24.27 

17.63 

24.19 

17.74 

24.12 

17.84 

24.04 

17.95 

30 

31 

25.08 

18.22 

25.00 

18.33 

24.92 

18.44 

24.84 

18.55 

31 

S2 

25.89 

18.81 

25.81 

18.92 

25.72 

19.03 

25.64 

19.15 

32 

33 

26.70 

19.40 

26.61 

19.51 

26.53 

19.63 

26.44 

19.74 

33 

34 

27.51 

19.98 

27.42 

20.10 

27.33 

20.22 

27.24 

20.34 

34 

35 

28.32 

20.57 

28.23 

20.70 

28.13 

20.82 

28.04 

20.94 

35 

36 

29.12 

21.16 

29.03 

21,29 

28.94 

21.41 

28.85 

21.54 

36 

37 

29.93 

21.75 

29.84 

21.88 

29.74 

22.01 

29  .  65 

22.14 

37 

38 

30.74 

22.34 

30.64 

22.47 

30.55 

22.60 

30.45 

22.74 

38 

39 

31.55 

22.92 

31.45 

23.06 

31.35 

23.20 

31.25 

23.33 

39 

40 

32.36 

23.51 

32.26 

23.65 

32.15 

23  .  79 

32.05 

23.93 

40 

41 

33.17 

24.10 

33.06 

24.24 

32.96 

24.39 

32.85 

24.53 

41 

42 

33.98 

24.69 

33.87 

24.83 

33  .  76 

24.98 

33.65 

25.13 

42 

43 

34.79 

25.27 

34.68 

25.43 

34.57 

25.58 

34  .'45 

25.73 

43 

44 

35.60 

25.86 

35.48 

26.02 

35.37 

26.17 

35.26 

26.33 

44 

45 

36.41 

26.45 

36.29 

26.61 

36.17 

26.77 

36.06 

26.92 

45 

46 

37.21 

27.04 

37.10 

27.20 

36.98 

27.36 

36.86 

27.52 

46 

47 

38.02 

27.63 

37.90 

27.79 

37.78 

27.96 

37.66 

28.12 

47 

48 

38.83 

28.21 

38.71 

28.38 

38.59 

28  .  55 

38,46 

28.72 

48 

49 

39.64 

28.80 

39.52 

28.97 

39.39 

29.15 

39.26 

29  .  32 

49 

50 

40.45 

29.39 

40.32 

29.57 

40.19 

29.74 

40.06 

29.92 

50 

a 

u 
c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

1 

CO 

b 

54  Deg. 

53|  Deg. 

53i  Deg. 

53i  Deg. 

3 

TRAVERSE    TABLE. 


75 


e 

36Deg. 

36i  Deg. 

36$  Deg. 

36|  Deg. 

O 

5.' 

P 

5' 

a 

g 

o 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

~51 

41.26 

29.98 

4l7l3~ 

30.16 

41.00 

30.34 

40.86 

30.51 

51 

52 

42.07 

30.56 

41.94 

30.75 

141.80 

30.93 

41.67 

31.11 

52 

53 

42.88 

31.15 

42.74 

31.34 

42.60 

31.53 

42.47 

31.71 

53 

54 

43.69 

31.74 

43.55 

31.93 

43.41 

32.12 

43.27 

32.31 

54 

55 

44.50 

32.33 

44.35 

32.52 

44.21 

32.72 

44.07 

32.91 

55 

56 

45.30 

32.92 

45.16 

33.11 

45.02 

33.31 

44.87 

33.51 

56 

57 

46.11 

33.50 

45.97 

33.70 

145.82 

33.90 

45.67 

34.10 

57 

58 

46.92 

34.09 

46.77 

34.30 

46.62 

34.50 

46.47 

34.70 

58 

59 

47.73 

34.68 

47.58 

34.89 

47.43 

35  09 

47.27 

35.30 

53 

60 

48.54 

35.27 

48.39 

35.48 

48.23 

35.69 

48.08 

35.90 

60 

61 

49.35 

35.85 

49.19 

36.07 

49.04 

36.28 

48.88 

36.50 

61 

62 

50.16 

36.44 

50.00 

36.66 

49.84 

36.88 

49.68 

37.10 

62 

63 

50.97 

37.03 

50.81 

37.25 

50.64 

37.47 

50.48 

37.69 

63 

64 

51.78 

37.62 

51.61 

37.84 

51.45 

38.07 

51.28 

38.29 

64 

65 

52.59 

38,21 

52.42 

38.44 

52.25 

38.66 

52.08 

38.89 

65 

66 

53.40 

38.79 

53.23 

39.03 

53.05 

39.26 

52.88 

39.49 

66 

67 

54.20 

39.38 

54.03 

39.62 

53.86 

39.85 

53.68 

40.09 

67 

68 

55.01 

39.97 

54.84 

40.21 

54.66 

40.45 

54.49 

40.69 

68 

69 

55.82 

40.56 

55.64 

40.80 

55.47 

41.04 

55.29 

41.28 

69 

70 

56.63 

41.14 

56.45 

41.39 

56.27 

41.64 

56.09 

41.88 

70 

71 

57.44 

41.73 

57.26 

41.98 

57.07 

42.23 

56.89 

42.48 

71 

72 

58.25 

42.32 

58.06 

42.57 

57.88 

42.83 

57.69 

43.08 

72 

73 

59.06 

42.91 

58.87 

43.17 

58.68 

43.42 

58.49 

43.68 

73 

74 

59.87 

43.50j 

59.68 

43.76 

59.49 

44.02 

59.29 

44.28 

74 

75 

60.68 

44.08 

60.48 

44.35 

60.29 

44.61 

60.09 

44.87 

75 

76 

61.49 

44.67 

61.29 

44.94 

61.09 

45.21 

60.90 

45.47 

76 

77 

62  .  29 

45.26 

62.10 

45.53 

61.90 

45.80 

61.70 

46.07 

77 

78 

63.10 

45.85 

62.90 

46.12 

62.70 

46.40 

62.50 

46.67 

78 

79 

63.91 

46.43 

63.71 

46.71 

63.50 

46.99 

63.30 

47.27 

79 

80 

64.72 

47.02! 

64.52 

47.30 

64.31 

47.59 

64.10 

47.87 

80 

81 

65.53 

47.61 

65.32 

47.90 

65.11 

48.18 

64.90 

48.46 

81 

82 

66.34 

48.20 

66.13 

48.49 

65.92 

48.78 

65.70 

49-06 

82 

83    67.15 

48.79 

66.93 

49.08 

66.72 

49.37 

66.50 

49.66 

83 

84 

67.96 

49.37 

67.74 

49.67 

67.52 

49.97 

67.31 

50.26 

84 

85 

68.77 

49.96 

68.55 

50.26 

68.33 

50.56 

68.11 

50.86 

85 

86 

69.58 

50.55 

69.35 

50.85 

69.13 

51.15 

68.91 

51.46 

86 

87 

70.38 

51.14 

70.16 

51.44 

69.94 

51.75 

69.71 

52.05 

87 

88 

71.19 

51.73 

70.97 

52.04 

70.74 

52.34 

70.51 

52.65 

88 

89 

72.00 

52.31 

71.77 

52.63 

71.54 

52.94 

71.31 

53.25 

89 

90    72.81 

52.90 

72.58 

53.22 

72.35 

53.53 

72.11 

53.85 

90 

91    73.62 

53.49! 

73.39 

53.81 

73.15 

54.13 

72.91 

54.45 

91 

92  174.43 

54.08 

74.19 

54.40 

73.95 

54.72 

73.72 

55.05 

92 

93  i  75.24 

54.66 

75.00 

54.99 

74.76 

55.32 

74.52 

55.64 

93 

94 

76.05 

55.25 

75.81 

55.58 

75.56 

55.91 

75.32 

50.24 

94 

95 

76.86 

55.84 

76.61 

56.17 

76.37' 

56.51 

76.12 

56.84 

95 

96 

77.67 

56.43 

77.42 

56.77 

77.17 

57.10 

76.92 

57.44 

96 

97 

78.47 

57.02 

78.23 

57.36 

77.97 

57.70 

77.72 

58.04 

97 

98 

79.28 

57.60 

79.03 

57.95 

78.78 

58.29 

78.52 

58.64 

98 

99    80.09 

58.19 

79.84 

58.54 

79.58 

58.89 

79.32 

59.23 

,99 

100  ,80.90 

58.78 

80.64 

59.13 

80.39 

59.48 

80.13 

59.83 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D*. 

Lat. 

od 

5 

54  Deg. 

53|  Deg. 

53i  Deg. 

53*  Deg. 

•£ 

- 

76 


TRAVERSE    TABLE. 


o 

37  Deg. 

37*  Deg. 

37*  Deg. 

37|  Deg. 

C 

en" 

^ 

3 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

f> 
S 
o 

1 

0.80 

0.601 

6.80 

0.61 

0.79 

0.61 

0.79 

0.61 

I 

2 

1.60 

1.20 

1.59 

1.21 

1.59 

1.22 

1.58 

1.22 

2 

3 

2.40 

1.81 

2.39 

1.82 

2.38 

1.83 

2.37 

1.84 

3 

4 

3.19 

2.41 

3.18 

2.42 

3.17 

2.43 

3.16 

2.45 

4 

5 

3.99 

3.01 

3.98 

3.03 

3.97 

3.04 

3.95 

3.06 

5 

6 

4.79 

3.61 

4.78 

3.63 

4.76 

3.65 

4.74 

3.67 

6 

7 

5.59 

4.21 

5.57 

4.24 

5.55 

4.26 

5.53 

4.29 

7 

8 

6.39 

4.81 

6.37 

4.84 

6.35 

4.87 

6.33 

4.90 

8 

9 

7.19 

5.42 

7.16 

5.45 

7.14 

5.48 

7.12 

5.51 

9 

10 

7.99 

6.02 

7.96 

6.05 

7.93 

6.09 

7.91 

6.12 

10 

.11 

8.78 

6.62 

8.76 

6.66 

8.73 

6.70 

8.70 

6.73 

11 

12 

9.58 

7.22 

9.55 

7.26 

9.52 

7.31 

9.49 

7.35 

12 

13 

10.38 

7.82 

10.35 

7.87 

10.31 

7.91 

10.28 

7.96 

13 

14 

11.18 

8.43 

11.14 

8.47 

11.11 

8.52 

11.07 

8.57 

14 

15 

11.98 

9.03 

11.94 

9.08 

11.90 

9.13 

11.86 

9.18 

15 

16 

12.78 

9.63 

12.74 

9.68 

12.69 

9.74 

12.65 

9.80 

16 

17 

13.58 

10.23 

13.53 

10.29 

13.49 

10.35 

13.44 

10.41 

17 

18 

14.38 

10.83 

14.33 

10.90 

14.28 

10.96 

14.23 

11.02 

18 

19 

15.17 

11.43 

15.12 

11.50 

15.07 

11.57 

15.02 

11.63 

19 

20 

15.97 

12.04 

15.92 

12.11 

15.87 

12.48 

15.81 

12.24 

20 

21 

16.77 

12.64 

16.72 

12.71 

16.66 

12.78 

16.60 

12.80 

21 

22 

17.57 

13.24 

17.51 

13.32 

17.45 

13.39 

17.40 

13.47 

22 

OO 

18.37 

13.84 

18.31 

13.92 

18.25 

14.00 

18.19 

14.08 

23 

24 

19.17 

14.44 

19.10 

14.53 

19.04 

14.61 

18.98 

14.69 

24 

25 

19.97 

15.05 

19.90 

15.13 

19.83 

15.22 

19.77 

15.31 

25 

26 

20.76 

15.65 

20.70 

15.74 

20.63 

15.83 

20.56 

15.92 

26 

27 

21.56 

16.25 

21.49 

16.34 

21.42 

16.44 

21.35 

16.53 

27 

28 

22.36 

16.85 

'22.29 

16.95 

22.21 

17.05 

22.14 

17.14 

28 

29 

23.16 

17.45 

23.08 

17.55 

23.01 

17.65 

22.93 

17.75 

29 

30 

23.96 

18.05 

23.88 

18.16 

23.80 

18.26 

23.72 

18.37 

30 

31 

24.76 

18.66 

24.68 

18.76 

24.59 

18.87 

24.51 

18.98 

31 

32 

25.56 

19.26 

25.47 

19.37 

25.39 

19.48 

25.30 

19.59 

32 

33 

26.35 

19.  8G 

26.27 

19.97 

26.18 

20.09 

26.09 

20.20 

33 

34 

27.15 

20.46 

27.06 

20.58 

26.97 

20.70 

26.88 

20.82 

34 

35 

27.95 

21.06 

27.86 

21.19 

27.77 

21.31 

27.67 

21.43 

35 

36 

28.75 

21.67 

28.66 

21.79 

28.56 

21.92 

28.46 

22.04 

36 

37 

29.55 

22.27 

29.45 

22.40 

29.35 

22.52 

29.26 

22.65 

37 

38 

30.35 

22.87 

30.25 

23.00 

30.15 

23.13 

30.05 

23.26 

38 

39 

31.15 

23.47 

31.04 

23.61 

30.94 

23.74 

30.84 

23.88 

39 

40 

31.95 

24.07 

31.84 

24.21 

31.73 

24,35 

31.63 

24.49 

40 

41 

32.74 

24.67 

32.64 

24.82 

32.53 

24.96 

32.42 

25.10 

41 

42 

33.54 

25.28 

33.43 

25.42 

33.32 

25.57 

33.21 

25.71 

4-2 

43 

34.34 

25.88 

34.23 

26.03 

34.11 

26.18 

34.00 

26.33 

43 

44 

35.14 

2C.48 

35.02 

26.63 

34.91 

26.79 

34.79 

26.94 

44 

45 

35.94 

27.08 

35.82 

27.24 

35.70 

27.39 

35.58 

27.55 

45 

46 

36.74 

27.68 

36.62 

27.84 

36.49 

28.00 

36.37 

28.16 

46 

47 

37.54 

28.29 

37.41 

28.45 

37.29 

28.61 

37.16 

28.77 

47 

48 

38.33 

28.89 

38.21 

29.05 

38.08 

29.22 

37.95 

29.39 

48 

49 

39.13 

29.49 

39.00 

29.66 

38.87 

29.83 

38.74 

30.00 

49 

50 

39.93 

30.09 

39.80 

30.26 

39.67 

30.44 

39.53 

30.61 

50 

g> 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

1 

£ 

53  Deg. 

52|  Deg. 

52i  Deg. 

52*  Deg. 

I 

TRAVERSE    TABLE. 


77 


o 

f 

37  Deg. 

37*  Deg. 

37  i  Deg. 

37|  Deg. 

C 

5- 

? 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

5 
P 

51 

40.73 

30.69 

40.60 

30.87 

40.46 

31.05 

40.33 

31.22 

51 

52 

41.53 

31.29 

41.39 

31.48 

41.25 

31.66 

41.12 

31.84 

52 

53 

42.33 

31.90 

42.19 

32.08 

42.05 

32.26 

41.91 

32.45 

53 

54 

43.13 

32.50 

42.98 

32.69 

42.84 

32.87 

42.70 

33.06 

54 

55 

43.92 

33.10 

43.78 

33.29 

43.63 

33.48 

43.49 

33.67 

55 

56 

44.72 

33.70 

44.58 

33.90 

44.43 

34.09 

44.28 

34.28 

56 

57 

45.52 

34.30 

45.37 

34.50 

45.22 

34.70 

45.07 

34.90 

57 

58 

46.32 

34.91 

46.17 

35.11 

46.01 

35.31 

45.86 

35.51 

58 

59 

47.12 

35.51 

46.96 

35.71 

46.81 

35.92 

46.65 

36.12 

59 

60 

47.92 

36.11 

47.76 

36.32 

47.60 

36.53 

47.44 

36.73 

60 

61 

48.72 

36.71 

48.56 

36.92 

48.39 

37.13 

48.23 

37.35 

61 

62 

49.52 

37.31 

49.35 

37.53 

49.19 

37.74 

49.02 

37.96 

62 

63 

50.31 

37.91 

50.15 

38.13 

49.98 

38.35 

49.81 

38.57 

63 

64 

51.11 

38.52 

50.94 

38.74 

50.77 

38.96 

50.60 

39.18 

64 

65 

51.91 

39.12 

51.74 

39.34 

51.57 

39.57 

51.39 

39.79 

65 

66 

52.71 

39.72 

52.54 

39.95 

52.36 

40.18 

52.19 

40.41 

66 

67 

53.51 

40.32 

53.33 

40.55 

53.15 

40.79 

52.98 

41.02 

67 

68 

54.31 

40.92 

54.13 

41.16 

53.95 

41.40 

53.77 

41.63 

68 

69 

55.11 

41.53 

54.92 

41.77 

54.74 

42.00 

54.56 

42.24 

69 

70 

55.90 

42.13 

55.72 

42.37 

55.53 

42.61 

55.35 

42  86 

70 

71 

56.70 

42.73 

56.52 

42.98 

56.33 

43.22 

56.14 

43.47 

71 

72 

57.50 

43.33 

57.31 

43.58 

57.12 

43.83 

56.93 

44.08 

72 

73 

58.30 

43.93 

58  .  1  1 

44.19 

57.91 

44.44 

57.72 

44.69 

73 

74 

59.10 

44.53 

58.90 

44.79 

58.71 

45.05 

58.51 

45.30 

74 

75 

59.90 

45.141 

59.70 

45.40 

59.50 

45.66 

59.30 

45.92 

75 

76 

60.70 

45.74 

60.50 

46.00 

60.29 

46.27 

60.09 

46.53 

76 

77 

61.49 

46.34 

61.29 

46.61 

61.09 

46.87 

60.88 

47.14 

77 

78 

62.29 

46.94 

62.09 

47.21 

61.88 

47.48 

61.67 

47.75 

78 

79 

63.09 

47.54 

62.88 

47.82 

62.67 

48.09 

62.46 

48.37 

79 

80 

63.89 

48.15 

63.68 

48.42 

63.47 

48.70 

63.20 

48.98 

80 

81 

64.69 

48.75 

64.48 

49.03 

64.26 

49.31 

64.05 

49.59 

81 

82 

65.49 

49.35 

65.27 

49.63 

65.05 

49.92 

64.84 

50.20 

82 

83 

66.29 

49.95 

66.07 

50.24 

6^.85 

50.53 

65.63 

50.81 

83 

84 

67.09 

50.55 

66.86 

50.84 

66.64 

51.14 

66.42 

51.43 

84 

85 

67.88 

51.15 

67.66 

51.45 

67.43 

51.74 

67.21 

52.04 

85 

86 

68.68 

51.76 

68.46 

52.06 

68.23 

52.35 

68.00 

52.65 

86 

87 

69.48 

52.36 

69.25 

52.66 

69.02 

52.96 

68.79 

53.26 

87 

88 

70.28 

52.96 

70.05 

53.27 

69.82 

53.57 

69.58 

53.88 

88 

89 

71.08 

53.56 

70.84 

53.87 

70.61 

54.18 

70.37 

54.49 

89 

90 

71.88 

54.16 

71.64 

54.48 

71.40 

54.79 

71.16 

55.10 

90 

91 

72.68 

54.77 

72.44 

55.08 

72.20 

55.40 

71.95 

55.  71 

91 

92 

73.47 

55.37 

73.23 

55.69 

72.99 

56.01 

72.74 

56.32 

92 

93 

74.27 

55.97 

74.03 

56.29 

73.78 

56.61 

73.53 

56.94 

93 

94 

75.07 

56.57 

74.82 

56.90 

74.58 

57.22 

74.32 

57.55 

94 

95 

75.87 

57.17 

75.62 

57.50 

75.37 

57.83 

75.12 

58.16 

95 

96 

76.67 

57.77 

76.42 

58.11 

76.16 

58.44 

75.91 

58.77 

96 

97 

77.47 

58.38 

77.21 

58.71 

76.96 

59.05 

76.70 

59.39 

97 

98 

78.27 

58,98 

78.01 

59.32 

77.75 

59.66 

77.49 

60.00 

98 

99 

79.06 

59.58 

78.80 

59.92 

78.54 

60.27 

78.28 

60.61 

99 

100  i  79.  86    60.18 

79.60 

60.53 

79.34 

60.88 

79.07 

61.22 

100 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| 

3 
5 
Q 

53  Deg. 

52|  Deg. 

52*  Deg. 

52|  Deg. 

5 

78 


TRAVJ1RSE    TABLE. 


.0 

38  Deg. 

38i  Deg. 

38i  Deg. 

381  Deg. 

O 

5* 
E 

55° 

go" 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  I  Dpp. 

Lat. 

Dep. 

3 
P 

i 

0.79 

0.62 

0.79 

0.62 

0.78 

0.62 

0.78 

0.63 

1 

2 

1.58 

1.23 

1.57 

1.24 

1.57 

1.24 

1.56 

1.25 

2 

3 

2.36 

1.85 

2.36 

•1.86 

2.35 

1.87 

2.34 

1.88 

3 

4 

3.15 

2.46 

3.14 

2.48 

3.13 

2.49 

3.12 

2.50 

4 

5 

3.94 

3.08 

3.93 

3.10 

3.91 

3.11 

3.90 

3.13 

5 

6 

4.73 

3.69 

4.71 

3.71 

4.70 

3.74 

4.68 

3.76 

6 

7 

5.52 

4.31 

5.50 

4.33 

5.48 

4.36 

5.46 

4.38 

7 

8 

6.30 

4.93 

6.28 

4.95 

6.26 

4.98 

6.24 

5.01 

8 

9 

tf.09 

5.54 

7.07 

5.57 

7.04 

5,60 

7.02 

5.63 

9 

10 

7.88 

6tl6 

7.85 

6.19 

7.83 

6.23 

7.80 

6.26 

10 

11 

8.67 

•6.77 

8.64 

6.81 

8.61 

6.85 

8.58 

6.89 

11 

12 

9.46 

7.39 

9.42 

7.43 

9.39 

7.47 

9.36 

7.51 

12 

13 

10.24 

8.00 

10.21 

8.05 

10.17 

8.09 

10.14 

8.14 

13 

14 

11.03 

8.62 

10.99 

8.67 

10.96 

8.72 

10.92 

8.76 

14 

15 

11.82 

9.23 

11.78 

9.29 

11.74 

9.34 

1  1  .  70 

9.39 

15 

16 

12.61 

9.85 

12.57 

9.91 

12.52 

9.96 

12.48 

10.01 

16 

17 

13.40 

10.47 

13.35 

10.52 

13.30 

10.58 

13.26 

10.64 

17 

18 

14.18 

11.08 

14.14 

11.14 

14.09 

11.21 

14.04 

11127 

18 

19    14.97 

11.70 

14.92 

11.76 

14.87 

11.83 

14.82 

11.89 

19 

20 

15.76 

12.31 

15.71 

12.38 

15.65 

12.45 

15.60 

12.52 

20 

21. 

16.55 

12.93 

16.49 

13.00 

16.43 

13.07 

16.38 

13.14 

21 

22 

17.34 

13.54 

17.28 

13.62 

17.22 

13.70 

17.16 

13.77 

22 

23 

18.12 

14.16 

18.06 

14.24 

18.00 

14.32 

17.94 

14.40 

23 

24 

18.91 

14.78 

18.85 

14.86 

18.78 

14.94 

18.72 

15.02 

24 

25 

19.70 

15.39 

19.63 

15.48 

19.57 

15.56 

19.50 

15.65 

25 

26 

20.49 

16.01 

20.42 

16.10 

20.35 

16.19 

20.28 

16.27 

26 

27 

21.28 

16.62 

21.20 

16.72 

21.13 

16.81 

21.06 

16.90 

27 

28 

22.06 

17.24 

21.99 

17.33 

21.91 

17.43 

21.84 

17.53 

28 

29 

22.85 

17.85 

22.77 

17.95 

22.70 

18.05 

22.62 

18.15 

29 

30 

23.64 

18.47 

23.56 

18.57 

23.48 

18.68 

23.40 

18.78 

30 

31 

24.43 

19.09 

24.34 

19.19 

24.26 

19.30 

24.18 

19.40 

31 

32 

25.22 

19.70 

25.13 

19.81 

25.04 

19.92 

24.96 

20.03 

32 

33 

26.00 

20.32 

25.92 

20.43 

25.83 

20.54 

25.74 

20.66 

33 

34 

26.79 

20.93 

26.70 

21.05 

26.61 

21.17 

26.52 

21.28 

34 

35 

27.58 

21.55 

27.49 

21.67 

27.39 

21.79 

27.30 

21.91 

35 

36 

28.37 

22.16 

28.27 

22  .  29 

28.17 

22.41 

28.08 

22.53 

36 

37 

29.16 

22.78 

29.06 

22.91 

28.96 

23.03 

28.86 

23.16 

37 

38 

29.94 

23.40 

29.84 

23  .  53 

29.74 

23.66 

29.64 

23.79 

38 

39 

30.73 

24.01 

30.63 

24.14 

30.52 

24.28 

30.42 

24.41 

39 

40 

31.52 

24.63 

31.41 

24.76 

31.30 

24.90 

31.20 

25.04 

40 

41 

32.31 

25.24 

32.20 

25.38 

32.09 

25  .  52 

31.98 

25.66 

41 

42 

33.10 

25.86 

32.98 

26.00 

32.87 

26.15 

32.76 

26.29 

42 

43 

33.88 

26.47 

33.77 

26.62 

33.65 

26.77 

33.53 

26.91 

4-3 

44 

34.67 

27.09 

34.55 

27.24 

34.43 

27.39 

34.31 

27.54 

44 

45 

•35.46 

27.70 

35.34 

27.86 

35.22 

28.01 

35.09 

28.17 

45 

46 

36.25 

28.32 

36.12 

28.48 

36.00 

28.64 

35.87 

28.79 

46 

47 

37.04 

28.94 

36.91 

29.10 

36.78 

29.26 

36.65 

29.42 

47 

48 

37.82 

29.55 

37.70 

29.72 

37.57 

29.88 

37.43 

30.04 

48 

49 

38.61 

30.17 

38.48 

30.34 

38.35 

30.50 

38.21 

30.67 

49 

50 

39.40 

30.78 

39.27 

30.95 

39.13 

31.13 

38.99 

31.30 

50 

B 
O 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

"co 

5 

52  Deg. 

51|  Deg. 

51i  Deg. 

51i  Deg. 

CO 

Q 

TRAVERSE    TABLE. 


70 


G 

38  Deg. 

38J  Deg. 

38i  Deg. 

38f  Deg. 

O 

P 

Lat. 

Dep. 

L^at. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

51 

40.19 

31.40 

40.05 

31.57 

39.91 

31.75 

39.77 

31.92 

TT 

52 

40.98 

32.01 

40.84 

32.19 

40.70 

32.37 

40.55 

32,55 

52 

53 

41.76 

32.63 

41.62 

32.81 

41.48 

32.99 

41.33 

33.17 

53 

54 

42.55 

33.25 

42.41 

33.43 

42.26 

33.62 

42.11 

33.80 

54 

55 

43.34 

33.86 

43.19 

34.05 

43.04 

34.24 

42.89 

34.43 

55 

56 

44.13 

34.48 

43.98 

34.67 

43.83 

34.86 

43.67 

35.05 

56 

57 

44.92 

35,09 

44.76 

35.29 

4-4.61 

35.48 

44.45 

35.68 

57 

58 

45  .  70 

3*^.71 

45.55 

35.91 

45.39 

36.11 

45.23 

36.30 

58 

59 

46.49 

36  .32 

46.33 

36.53 

46.17 

36  .  73 

46.01 

36.93 

59 

60 

47.28 

36.94 

47*12 

37.15 

46.96 

37.35 

46.79 

37.56 

60 

61 

48.07 

37.56 

47.90 

37.76 

47.74 

37.97 

47.57 

38.18 

61 

62 

48.86 

38.17 

48.69 

38.38 

48.52 

38.60 

48.35 

38.81 

62 

63 

49.64 

38.79 

49.47 

39.00 

49.30 

39.22 

.49.13 

39.43 

63 

64 

50.43 

39.40 

50.26 

39.62 

50.09 

39.84 

49.91 

40.06 

64 

65 

51.22 

40.02 

51.05 

40.24 

50.87 

40.46 

50.69 

40.68 

65 

66 

52.01 

40.63 

51.83 

40.86 

51.65 

41.09 

51.47 

41.31 

66 

67 

52.80 

41.25 

52.62 

41.48 

52.43 

41.71 

52.25 

41.94 

67 

68 

53.58 

41.86 

53.40 

42.10 

53.22 

42.33 

53.03 

42.56 

68 

69 

54.37 

42.48 

54.19 

42.72 

54.00 

42.95 

53.81 

43.  19 

69 

70 

55.16 

43.10 

54.97 

43.34 

54.78 

43.58 

54.59 

43.81 

70 

71 

55.95 

43.71 

55.76 

43.96 

55.57 

44.20 

55.37 

44.44 

71 

72 

56.74 

44.33 

56.54 

44.57 

56.35 

44.82 

56.15 

45.07 

72 

73 

57.52 

44.94 

57.33 

45.19 

57.13 

45.44 

56.93 

45.69 

73 

74 

58.31 

45.56 

58.11 

45.81 

57.91 

46.07 

57.71 

46.32 

74 

75 

59.10 

46.17 

58.90 

46.43 

58.70 

46.69 

58.49 

46.94 

75 

76 

59.89 

46.79 

59.68 

47.05 

59.48 

47,31 

59.27 

47.57 

76 

77 

60.68 

47.41 

60.47 

47.67 

60.26 

47,93 

60.05 

48.20 

77 

78 

61.46 

48.02 

61.25 

48.29 

61.04 

48.56 

60.83 

48.82 

78 

79 

62.25 

48.64 

62.04 

48.91 

61.83 

49.18 

61.61 

49.45 

79 

80 

63.04 

49.25 

62.83 

49.53 

62.61 

49.80 

62.39 

50.07 

80 

81 

63.83 

49.87 

63.61 

50.15 

63.39 

50.42 

63.17 

50.70 

81 

82 

64.62 

50.48 

64.40 

50.77 

64.17 

51.05 

63.95 

51.33 

82 

83 

65.40 

51.10 

65.18 

51.38 

64.96 

51.67 

64.73 

51.95 

83 

84 

66.19 

51.72 

65.97 

52.00 

65.74 

52.29 

65.51 

52.58 

84 

85 

66.98 

52.33 

66.75 

52.62  i 

66.52 

52.91 

66.29 

53.20 

85 

86 

67.77 

52.95 

67.54 

53.^4! 

67.30 

53.54 

67.07 

53.83 

86 

87 

68.56 

53.56 

68.32 

53.86  ! 

68.09 

54.16 

67.85 

54.46 

87 

88 

69.34 

54.18 

69.11 

54.48  i 

68.87 

54.78 

68.63 

55.08 

88 

89 

70.13 

54.79 

69.89 

55.10  ; 

69.65 

55.40 

69.41 

55.71 

89 

90 

70.92 

55.41 

70.68 

55  .  72 

70.43 

56.03 

70.19 

56.33 

90 

91 

71.71 

56.03 

71.46 

56.34 

71.22 

56.65 

70.97 

56.96 

91 

92 

72.50 

56.64 

72.25 

56.96 

72.00 

57.27 

71.75 

57.58 

92 

93 

73.28 

57.26 

73.03 

57.58  ; 

72.78 

57.89 

72.53 

58.21 

93 

94 

74.07 

57.87 

73.82 

58.19 

73.57 

58.52 

73.31 

58.84 

94 

95 

74.86 

58.49 

74.61 

58.81 

74.35 

59.14 

74.09 

59.46 

95 

96 

75.65 

59.10 

75.39 

59.43     75.13 

59.76 

74.87 

60.09 

96 

97 

76.44 

59.72 

76.18 

60.05     75.91 

60.  3S 

75.65 

60.71 

97 

98 

77.22 

60.33 

76.96 

60.67  ,76.70 

61.01 

76.43 

61.34 

98 

99 

78.01 

60.95 

77.75 

61.29     77.48 

61.63 

77.21 

61.97 

99 

100 

78.80 

61.57 

78.53 

61.91  J78.26 

62.25 

77.99 

62.59 

100 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

a 

52  Deg. 

\ 
51J  Deg.           51fDeg. 

5U  Deg. 

1 

5 

1 

TRAVERSE   TABLE. 


o 

? 

39  Deg. 

39*  Deg. 

39£  Deg. 

39|  Deg. 

O 
£' 

CJ 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1 

0.78 

0.63 

0.77 

0.63 

0.77 

0.64 

0.77 

0.64 

] 

2 

1.55 

1.26 

1.55 

1.27 

1.54 

1.27 

1.54 

1.28 

f 

3 

2.33 

1.89 

2.32 

1.90 

2.31 

1.91 

2.31 

1,92 

f. 

4 

3.11 

2.52 

3.10 

2.53 

3.09 

2.54 

3.08 

2.56 

i 

5 

3.89 

3.15 

3.87 

3.16 

3.86 

3.18 

3.84 

3.20 

5 

6 

4.66 

3.78 

4.65 

3.80 

4.63 

3.82 

4.61 

3.84 

6 

7 

5.44 

4.41 

5.42 

4.43 

5.40 

4.45 

5.38 

4.48 

7 

8 

6.22 

5.03 

6.20 

5.06 

6.17 

5.09 

6.1'5 

5.12 

8 

9 

6.99 

5.66 

6.97 

5.69 

6.94 

5.72 

6.92 

5.75 

9 

10 

7.77 

6.29 

7.74 

6.33 

7.72 

6.36 

7.69 

6.39 

10 

11 

8.55 

6.92 

8.52 

6.96 

8.49 

7.00 

8.46 

7.03 

11 

12 

9.33 

7.55 

9.29 

7.59 

9.26 

7.63 

9.23 

7.67 

12 

13 

10.10 

8.18 

10.07 

8.23 

10.03 

8.27 

9.99 

8.31 

13 

14 

10.88 

8.81 

10.84 

8.86 

10.80 

8.91 

10.76 

8.95 

14 

15 

11.66 

9.44 

11.62 

9.49 

11.57 

9.54 

11.53 

9.59 

15 

16 

12.43 

10.07 

12.39 

10.12 

12.35 

10.18 

12.30 

10.23 

16 

17 

13.21 

10.70 

13.16 

10.76 

13.12 

10.81 

13.07 

10.87 

17 

18 

13.99 

11.33 

13.94 

11.39 

13.89 

11.45 

13.84 

11.51 

18 

19 

14.77 

11.96 

14.71 

12.02 

14.66 

12.09 

14.61 

12.15 

19 

20 

15.54 

12.59 

15.49 

12.65 

15.43 

12.72 

15.38 

12.79 

20 

21 

16.32 

13.22 

16.26 

13.29 

16.20 

13.36 

16.15 

13.43 

21 

22 

17.10 

13.84 

17.04 

13.92 

16.98 

13.99 

16.91 

14.07 

22 

23 

17.87 

14.47 

17.81 

14.55 

17.75 

14.63 

17.68 

14.71 

23 

24 

18.65 

15.10 

18.59 

15.18 

18.52 

15.27 

18.45 

15.35 

24 

25 

19.43 

15.73 

19.36 

15.82 

19.29 

15.90 

19.22 

15.99 

25 

28 

20.21 

16.36 

20.13 

16.45 

20.06 

16.54 

19.99 

16.63 

26 

27 

20.98 

16.99 

20.91 

17.08 

20.83 

17.17 

20.76 

17.26 

27 

29 

21.76 

17.62 

21.68 

17.72 

21.61 

17.81 

21.53 

17.90 

28 

29 

22.54 

18.25 

22.46 

18.35 

22.38 

18.45 

22;  30 

18.54 

29 

30 

23.31 

18.88 

23.23 

18.98 

23.15 

19.08 

23.07 

19.18 

30 

31 

24.09 

19.51 

24.01 

19.61 

23.92 

19.72 

23.83 

19.82 

31 

32 

24.87 

20.14 

24.78 

20.25 

24.69 

20.35 

24.60 

20.46 

32 

33 

25.65 

20.77 

25.55 

20.88 

25.46 

20.99 

25.37 

21.10 

33 

34 

26.42 

21.40 

26.33 

21.51 

26.24 

21.63 

26.14 

21.74 

34 

35 

27.20 

22.03 

27.10 

22.14 

27.01 

22.26 

26.91 

22  .  38 

35 

36 

27.98 

22.66 

27.88 

22.78 

27.78 

22.90 

27.68 

23.02 

36 

37 

28.75 

23.28 

28.65 

23.41 

28.55 

23.53 

28.45 

23.66 

37 

38 

29.53 

23.91 

29.43 

24.04 

29.32 

24.17 

29.22 

24.30 

38 

39 

30.31 

24.54 

30.20 

24.68 

30.09 

24.81 

29.98 

24.94 

39 

40 

31.09 

25.17 

30.98 

25.31 

30.86 

25.44 

30  .  75 

25.58 

40 

41 

31.86 

25,80 

31.75 

25.94 

31.64 

26.08 

31.52 

26.22 

41 

42 

32.64 

26.43 

32.52 

26.57 

32.41 

26.72 

32.29 

26.86 

42 

43 

33.42 

27.06 

33.30 

27.21 

33.18 

27.35 

33.06 

27.50 

43 

44 

34.19 

27.69 

34.07 

27.84 

33.95 

27.99 

33.83 

28.14 

44 

45 

34.97 

28.32 

34.85 

28.47 

34.72 

28.62 

34.60 

28.77 

45 

46 

35.75 

28.95 

35.62 

29.10 

35.49 

29.26 

35.37 

29.41 

46 

47 

36.53 

29.58 

36.40 

29.74 

36.27 

29.90 

36.14 

30.05 

47 

48 

37.30 

B0»21 

37.17 

30  .  37 

37.04 

30.53 

36.90 

30.69 

48 

49 

38.08 

30.84 

37.95 

31.00 

37.81 

31.17 

37.67 

31.33 

49 

50 

38.86 

31.47 

38.72 

31.64 

38.58 

31.80 

38.44 

31.97 

50 

8 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

£ 

£ 

to 

$ 

.  Q 

51  Deg. 

50|  Deg. 

50i  Dog. 

50i  Deg. 

s 

TRAVERSE    TABLE. 


G 

59  Deg. 

.39*  Deg. 

39*  Deg. 

39}  Deg. 

g 

£• 

1 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 
8 

51 

39.63 

32.10 

39.49 

32.27 

39.35 

32.44 

39.21 

32.61 

51 

52 

40.41 

32.72 

40.27 

32.90 

40.12 

33.08 

39.98 

33.25 

52 

53 

41.19 

33.35 

41.04 

33.53 

40.90 

33.71 

40.75 

33.89 

53 

54 

41.97 

33.98 

41.82 

34.17 

41.67 

34.35 

41.52 

34.53 

54 

60 

42.74 

34.61 

42.59 

34.80 

42.44 

34.98 

42.29 

35.17 

55 

56 

43.52 

35.24 

43.37 

35.43 

43.21 

35.62 

43.06 

35.81 

56 

57 

44.30 

35.87 

44.14 

36.06 

43.98 

36.26 

43.82 

36.45 

57 

58 

45.07 

36.50 

44.91 

36.70 

44.75 

36.89 

44.59 

37.09 

58 

59 

45.85 

37.13 

45.69 

P7.33 

45.53 

37.53 

45.36 

37.73 

59 

60 

46.63 

37.76 

46.46 

37.96 

46.30 

38.16 

46.13 

38.37 

60 

61 

47.41 

38.39 

47.24 

38.60 

47.07 

38.80 

46.90 

39.01 

61 

62 

48.18 

39.02 

48.01 

39.23 

47.84 

39.44 

47.67 

39.65 

62 

63 

48.96 

39.65 

48.79 

39.86 

48.61 

40.07 

48.44 

40.28 

63 

64 

49.74 

40.28 

49.56 

40.49 

49.38 

40.71 

49.21 

40.92 

64 

65 

50.51 

40.91 

50.34 

,41.13 

50.16 

41.35 

49.97 

41.56 

65 

66 

51.29 

41.54 

51.11 

41.76 

50.93 

41.98 

50.74 

42.20 

66 

67 

52.07 

42.  re 

51.88 

42.39 

51.70 

42.62 

51.51 

42.84 

67 

68 

52.85 

42.79 

52.66 

43.02 

52.47 

43.25 

52.28 

43.48 

68 

69 

53.52 

43.42 

53.43 

43.66 

53.24 

43.89 

53.05 

44.12 

69 

70 

54.40 

44.05 

54.21 

44.29 

54.01 

44.53 

53.82 

44.76 

70 

71 

55.18 

44.68 

54.98 

44.92 

54.79 

45.16 

54.59 

45.40 

71 

72 

55.95 

45.31 

55.76 

45.55 

55.56 

45.80 

55.36 

46.04 

72 

73 

56.73 

45.94 

56.53 

46.19 

56.33 

46.43 

56.13 

46.68 

73 

74 

57.51 

46.57 

57.31 

46.82 

57.10 

47.07 

56.89 

47.32 

74 

75 

58.29 

47.20 

58.08 

47.45 

57.87 

47.71 

57.66 

47.96 

75 

76 

59.06 

47.83 

58.85 

48.09 

58.64 

48.34 

58.43 

48.60 

76 

77 

59.84 

48.46 

59.63 

48.72 

59.42 

48.98 

59.20 

49.24 

77 

78 

60.62 

49.09 

60.40 

49.35 

60.19 

49.61 

59.97 

49.88 

78 

79 

61.39 

49.72 

61.18 

49.98 

60.96 

50.25 

60.74 

50.52 

79 

80 

62.17 

50.35 

61.95 

50.62 

61.73 

50.89 

61.51 

51.16 

80 

81 

62.95 

50.97 

62.73 

51.25 

62.50 

51.52 

62.28 

51.79 

81 

82 

63.73 

51.60 

63.50 

51.88 

63.27 

52.16 

63.04 

52.43 

82 

83 

64.50 

52.23 

64.27 

52.51 

64.04 

52.79 

63.81 

53.07 

83 

84 

65.28 

52.86 

65.05 

53.15 

64.82 

53.43 

64.58 

53.71 

84 

85 

66.06 

53.49 

65.82 

53.78 

65.59 

54.07 

65.35 

54.35 

85 

86 

66.83 

54.12 

66.60 

54.41 

66.36 

54.70 

66.12 

54.99 

86 

87 

67.61 

54.75 

er.37 

55.05 

67.13 

55.34 

66.89 

55.63 

87 

88 

68.39 

55.38 

68.15 

55.68 

67.90 

55.97 

67.66 

56.27 

88 

89 

69.17 

56.01 

68.92 

56.32 

68.67 

56.61 

68.43 

56.91 

89 

90 

69.94 

56.64 

69.70 

56.94 

69.45 

57.25 

69.20 

57.55 

90 

91 

70.72 

57.27 

70.47 

57.58 

70.22 

57.88 

69.96 

58.19 

91 

92 

71.50 

57.90 

71.24 

58.21 

70.99 

58.52 

70.73 

58.83 

92 

93 

72.27 

58.53 

72.02 

58.84 

71.76 

59.16 

71.50 

59.47 

93 

94 

73.05 

59.16 

72.79 

59.47 

72.53 

59.79 

72.27 

60.11 

94 

95 

73.83 

59.79 

73.57 

60.11 

73.30 

60.43 

73.04 

60.75 

95 

96 

74.61 

60.41 

74.34 

60.74 

74.08 

61.06 

73.81 

61.39 

96 

97 

75.38 

61.04 

75.12 

61.37 

74.85 

61.70 

74.58 

62.03 

97 

98 

76.16 

61.67 

75.89 

62.01 

75.62 

62.34 

75.35 

62.66 

98 

99 

76.94 

62.30 

76.66 

62.64 

76.39 

62.97 

76.12 

63.30 

99 

100 

77.71 

62.93 

77.44 

63.27 

77.16 

63.61 

76.88 

63.94 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

1 

51  Deg. 

50}  Deg. 

50i  Deg. 

50i  Deg. 

5 

TRAVEftSE   TABLE. 


o 

40  Deg. 

40}  Deg. 

40i  Deg. 

40|  Deg. 

5 

B 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

1 

0.77 

0.64 

0.76 

0,65 

0.76 

0.65 

0.76 

0.65 

i 

2 

1.53 

1.29 

1.53 

1.29 

1.52 

1.30 

1.52 

1.31 

2 

3 

2.30 

1.93 

2.29 

1.94 

2.28 

1.95 

2.27 

1.96 

3 

4 

3\06 

2.57 

3.05 

2.58 

3.04 

2.60 

3.03 

8.61 

4 

5 

3.83 

3.21 

3.82 

3.23 

3.80 

3.25 

3.79 

3.26 

5 

6 

4.60 

3.86 

4.58 

3.88 

4.56 

3.90 

4.55 

3.92 

6 

7 

5.36 

4.50 

5.34 

4.52 

5.32 

4.55 

5.30 

4.57 

7 

8 

6.13 

6.14 

6.11 

5.17 

6.08 

5.20 

6.06 

5.22 

8 

9 

6.89 

5.79 

6.87 

5.82 

6.84 

5.84 

6.82 

5.87 

9 

10 

7.66 

6.43 

7.63 

6.46 

7.60 

6.49 

7.58 

6.53 

10 

11 

8.43 

7.07 

8.40 

7.11 

8.36 

7.14 

8.33 

7.18 

11 

12 

9.19 

7.71 

9.16 

7.75 

9.12 

7.79 

9.09 

7.83 

12 

13 

9.96 

8.36 

9.92 

8.40 

9.89 

8.44 

9.85 

8.49 

13 

14 

10.72 

9.00 

10.69 

9.05 

10.65 

9.09 

10.61 

9.14 

14 

15 

11.49 

9.64 

11.45 

9.69 

11.41 

•9.74 

11.36 

9.79 

15 

16 

12.26 

10.28 

12.21 

10.34 

12.17 

10.39 

12.12 

10.44 

16 

17 

13.02 

10.93 

12.97 

10.98 

12.93 

11.04 

12.88 

11.10 

17 

18 

13.79 

11.57 

13.74 

11.63 

13.69 

11.69 

13.64 

11.75 

18 

19 

14.55 

12.21 

14.50 

12.28 

14.45 

12.34 

14.39 

12.40 

19 

20 

15.32 

12.86 

15.26 

12.92 

15.21 

12.99 

15.15 

13.06 

20 

21 

16.09 

13.50 

16.03 

13.57 

15.97 

13.64 

15.91 

13.71 

21 

22 

16.85 

14.14 

16.79 

14.21 

16.73 

14.29 

16.67 

14.36 

22 

23 

17.62 

14.78 

17.55 

14.86 

17.49 

14.94 

17.42 

15.01 

23 

24 

18.39 

15.43 

18.32 

15.51 

18.25 

15.59 

18.18 

15,67 

24 

25 

19.15 

16.07 

19.08 

16.15 

19.01 

16.24 

18.94 

16.32 

25 

26 

19.92 

16.71 

19.84 

16.80 

19.77 

16.89 

19.70 

16.97 

26 

27 

20.68 

17.36 

20.61 

17.45 

20.53 

17.54 

20.45 

17.62 

27 

28 

21.45 

18.00 

21.37 

18.09 

21.29 

18.18 

21.21 

18.28 

28 

29 

22.22 

18.64 

22.13 

18.74 

22.05 

18.83 

21.97 

18.93 

29 

30 

22.98 

19.28 

22.90 

19.38 

22.81 

19.48 

22.73 

19.58 

30 

31 

23.75 

19.93 

23.66 

20.03 

23.57 

20.13 

23.48 

20.24 

31 

32 

24.51 

20.57 

24.42 

20.68 

24.33 

20.78 

24.24 

20.89 

32 

33 

25.28 

21.21 

25.19 

21.32 

25.09 

21.43 

25.00 

21.54 

33 

34 

26.05 

21.85 

25.95 

21.97 

25.85 

22.08 

25.76 

22.19 

34 

35 

26.81 

22.50 

26.71 

22.61 

26.61 

22.73 

26.51 

22.85 

35 

36 

27.58 

23.14 

27.48 

23.26 

27.37 

23.38 

27.27 

23.50 

36 

37 

28.34 

23.78 

28.24 

23.91 

28.13 

24.03 

28.03 

24.15 

37 

38 

29.11 

24.43 

29.00 

24.55 

28.90 

24.68 

28.79 

24.80 

38 

39 

29.88 

25.07 

29.77 

25  .  20 

29.66 

25.33 

29.54 

25.46 

39 

40 

30.64 

25.71 

30.53 

25.84 

30.42 

25.98 

30,30 

26.11 

40 

41 

31.41 

26.35 

31.29 

26.49 

31.18 

26.03 

3i;oe 

26.76 

41 

42 

32.17 

27.00 

32.06 

27.14 

31.94 

27.28 

31.82 

27.42 

42 

43 

32.94 

27.64 

32.82 

27.78 

32.70 

27.93 

32.58 

28.07 

43 

44 

33.71 

28.28 

33.58 

28.43 

33.46 

28.58 

33.33 

28.72 

44 

45 

34.47 

28.93 

34.35 

29.08 

34.22 

29.23 

34.09 

29.37 

45 

46 

35.24 

29.57 

35.11 

29.72 

34.98 

29.87 

34.85 

30.03 

46 

47 

36.00 

30.21 

35.87 

30.37 

35.74 

30.52 

35.61 

30.68 

47 

48 

36.77 

30.85 

36.64 

31.01 

36.50 

31.17 

36.36 

31.33 

48 

49 

37.54 

31.50 

37.40 

31.66 

37.26 

31.82 

37.12 

31.99 

49 

50 

38.30 

32.14 

38.16 

32.31 

38.02 

32.47 

37.88 

32.64 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i 

5 

•^ 

•2 

o 

50  Deg. 

49|  Deg. 

49i  Deg. 

*  49i  Deg. 

Q 

TRAVERSE    TABLE. 


03 


Cj 

40  Deg. 

401  Deg. 

404  Deg. 

40|  Deg. 

5 

3 
O 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

51 

39.07 

32.78 

38.92 

32.95 

38  .  78 

33.12 

38.64 

33.29 

51 

52 

39.83 

33.42 

|39.69 

33.60 

39.54 

33.77 

39.39 

33.94 

52 

53 

40.60 

34.07 

40.45 

34.34 

40.30 

34.42 

40.15 

34.60 

53 

54 

41.37 

84.71 

41.21 

34.89 

41.06 

35.07 

40.91 

35.25 

54 

55 

42.13 

35.35 

41.98 

35  .  54 

41.82 

35.72 

41.67 

35.90 

55 

56 

42  .  90 

36.00 

42.74 

36.18 

42.58 

36.37 

42.42 

36.55 

56 

57 

43  .  66 

36.64 

43  .  50 

36.83 

43.34 

37.02 

43.18 

37.21 

57 

58 

44.43 

37.28 

44.27 

37.48 

44.10 

37.67 

43.94 

37.86 

58 

59 

45.20 

37.92 

45.03 

38.12 

44.86 

38.32 

44.70 

38.51 

59 

60 

45.96 

38.57 

45  .  79 

38.77 

45.62 

38.97 

45.45 

39.17 

60 

61 

46.73 

39.21 

46.56 

39.41 

46.38 

39.62 

46.21 

39.82 

61 

62 

47.49 

39.85 

47.32 

40.06 

47.15 

40.27 

46.97 

40.47 

62 

63 

48.26 

40.50 

48.08 

40.71 

47.91 

40.92 

47.73 

41.12 

63 

64 

49.03 

41.14 

48.85 

41.35 

48.67 

41.56 

48.48 

41.78 

64 

60 

49  .  79 

41.78 

49.61 

42.00 

49.43 

42.21 

49.24 

42.43 

65 

66 

50.56 

42.42 

50.37 

42.64 

50.19 

42.86 

50.00 

43.08 

66 

67 

51.32 

43.07 

51.14 

43.29 

50.95 

43.51 

50.76' 

43.73 

67 

63 

52.09 

43.71 

51.90 

43.94 

51.71 

44.16 

51.51 

44.39 

68 

69 

52.86 

44.35 

52  .  66 

44.58 

52.47 

44.81 

52.27 

45.04 

69 

70 

53.62 

45.00 

53.43 

45.23 

53.23 

45.46 

53.03 

45.69 

70 

71 

54.39 

45.64 

54.19 

45.87 

53.99 

46.11 

53.79 

46.35 

71 

72 

55.16 

46.28 

54.95 

46.52 

54.75 

46.76 

54.54 

47.00 

72 

73 

55  .  92 

46.92 

55.72 

47.17 

55.51 

47.41 

55.30 

47.65 

73 

T4 

56.69 

47.57 

56.48 

47.81 

56.27 

48.06 

56.06 

48.30 

74 

75 

57.45 

48.21 

57.24 

48.46: 

57.03 

48.71  ! 

56.82 

48.96 

75 

76 

58.22 

48.85 

58.01 

49.11  1 

57.79 

49.36  i 

57.57 

49.61 

76 

77 

58  .  99 

49.49 

58.77 

49.75 

58.55 

50.01  1 

58  .  33 

50.26 

77 

78 

59  .  75 

50.14 

59,53 

50.40 

59.31 

50.66  i 

59.09 

50.92 

78 

79 

60.52 

50.78 

60.30 

51.04 

60.07 

51.31 

59.85 

51.57 

79 

80 

61.28 

51.42 

61.06 

51.69 

60.83 

51.96 

60.61 

52.22 

80 

81 

62.05 

52.07 

-61.82 

52.341 

61.59 

52.61 

61.36 

52.87 

81 

82 

62.82 

52.71 

62.59 

52.98 

62.35 

53.25 

62.12 

53  .  53 

82 

83 

63.58 

53.35 

63.35 

53.63 

63.11 

53.90 

62.88 

54.18 

83 

84 

64.35 

53.99 

64.11 

54.27 

63.87 

54.55 

63.64 

54.83 

84 

85 

65.11 

54.64 

64.87 

54.92 

64.63 

55.20 

64.39 

55.48     85 

86 

65.88 

55.28 

65.64 

55.57 

65.39 

55.85 

65.15 

56.14 

86 

87 

66.65 

55.92 

66.40 

56.21 

66.16 

56.50 

65.91 

56.79 

87 

88 

67.41 

56.57 

67.16 

56.86 

66.92 

57.15 

66.67 

57.44 

88 

89 

68.18 

57.21 

67.93 

57.50 

67.68 

57.80 

67.42 

58.10 

89 

90 

68.94 

57.85 

68.69' 

58.15 

68.44 

58.45 

68.18 

58  .  75 

90 

91 

69.71 

58.49 

69.45 

58.80  j 

69.20 

59.10 

68.94 

59.40 

91 

92 

70.48 

59.14 

70.22 

59.44 

69.96 

59.75 

69.70 

60.05 

92 

93 

71.24 

59  .  78 

70.98 

60.09; 

70.72 

60.40 

70.45 

60.71 

93 

94 

72.01 

60.42 

71.74 

60.74: 

71.48 

61.05 

71.21 

61.36 

94 

95 

72.77 

61.06 

72.51 

61.38 

72.24 

6,1.70 

71.97 

62.01 

95 

96 

73.54 

61.71 

73.27 

62.03 

73.00 

62.35! 

72.73 

62.66 

96 

97 

74.31 

62.35 

74.03 

62.67 

73.76 

63.00 

73.48 

63.32 

97 

98 

75.07 

62.99 

74.80 

63.32 

74.52 

63.65 

74.24 

63.97 

98 

99 

75.84 

63.64 

75.56 

63.97 

75.28 

64.30 

75.00 

64.62 

99 

100 

76.60 

64.28 

76.32 

64.61 

76.04 

64.94 

75.76 

65.28 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

s 

a 

Q 

50  Deg. 

49f  Deg. 

494  Deg. 

49i  Deg. 

P 

84 


TRAVERSE    TABLE. 


5" 

41  Deg. 

4U  Deg. 

41|  Deg. 

4,|  D* 

0 

p 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o 

0> 

1 

0.75 

0.66 

0.75 

O.C6 

0.75 

0.66 

0.75 

0.67 

J 

2 

1.51 

1.31 

1.50 

1.32 

1.50 

1.33 

1.49 

1.33 

2 

3 

2.26 

1.97 

2.26 

1.98 

2.25 

1.99 

2.24 

2.00 

3 

4 

3.02 

2.62 

3.01 

2.64 

3.00 

2.65 

2.98 

2.66 

4 

5 

3.77 

3.28 

3.76 

3.30 

3.74 

3.31 

3.73 

3.33 

5 

6 

4.53 

3.94 

4.51 

3.96 

4.49 

3.98 

4.48 

4.00 

6 

7 

5.28 

4.59 

5.26 

4.62 

5.24 

4.64 

5.22 

4.66 

7 

8 

6.04 

5.25 

6.01 

5.27 

5.99 

5.30 

5.97 

5.33 

8 

9 

6.79 

5.90 

6.77 

5.93 

6.74 

5.96 

6.71 

5.99 

9 

10 

7.55 

6.56 

7.52 

6.59 

7.49 

6.63 

7.46 

6.66 

10 

11 

8.30 

7.22 

8.27 

7.25 

8.24 

7.29 

8.21 

7.32 

11 

12 

9.06 

7.87 

9.02 

7.91 

8.99 

7.95 

8.95 

7.99 

12 

13 

9.81 

8.53 

9.77 

8.57 

9.74 

8.61 

9.70 

8.66 

13 

14 

10.57 

9.18 

10.53 

9.23 

10.49 

9.28 

10.44 

9.32 

14 

15 

11.32 

9.84 

11.28 

9.89 

11.23 

9.94 

11.19 

9.99 

15 

16 

12.08 

10.50 

12.03 

10.55 

11.98 

10.60 

11.94 

10.65 

16 

17 

12.83 

11.15 

12.78 

11.21 

12.73 

11.26 

12.68 

11.32 

17 

18 

13.58 

11.81 

13.53 

11.87 

13.48 

11.93 

13.43 

11.99 

18 

19 

14.34 

12.47 

14.28 

12.53 

14.23 

12.59 

14.18 

12.65 

19 

20 

15.09 

13.12 

15.04 

13.19 

14.98 

13.25 

14.92 

13.32 

20 

21 

15.85 

13.78 

15.79 

13.85 

15.73 

13.91 

15.67 

13.98 

21 

22 

16.60 

14.43 

16.54 

14.51 

16.48 

14.58 

16.41 

14.65 

22 

23 

17.36 

15.09 

17.29 

15.16 

17.23 

15.24 

17.16 

15.32 

23 

24 

18.11 

15.75 

18.04 

15.82 

17.97 

15.90 

17.91 

15.98 

24 

25 

18.87 

16.40 

18.80 

16.48 

18.72 

16.57 

18.65 

16.65 

25 

26 

19.62 

17.06 

19.55 

17.14 

19.47 

17.23 

19.40 

17.31 

26 

27 

20.38 

17.71 

20.30 

17.80 

20.22 

17.89 

20.14 

17.98 

27 

28 

21.13 

18.37 

21.05 

18.46 

20.97 

18.55 

20.89 

18.64 

28 

29 

21.89 

19.03 

21.80 

19.12 

21.72 

19.22 

21.64 

19.31 

29 

30 

22.64 

19.68 

22.56 

19.78 

22.47 

19.88 

22.38 

19.98 

30 

31 

23.40  120.34 

23.31 

20.44 

23.22 

20.54 

23.13 

20.64 

31 

32 

24.15  120.99 

24.06 

21.10 

23.97 

21.20 

23.87 

21.31 

32 

33 

24.91    21.65 

24.81 

21.76 

24.72 

21.87 

24.62 

21.97 

33 

34 

25.66 

22.31 

25.56 

22.42 

25.46 

22.53 

25.37 

22.64 

34 

35 

26.41 

22.96 

26.31 

23.08 

26.21 

23.19 

26.11 

23.31 

35 

36 

27.17 

23.62 

27.07 

23.74 

26.96 

23.85 

26.86 

23.97 

36 

37 

27.92 

24.27 

27.82 

24.40 

27.71 

24.52 

27.60 

24.64 

37 

38 

28.68 

24.93 

28.57 

25.06 

28.46 

25.18 

28  .  35 

25.30 

38 

39 

29.43 

25.59 

29.32 

25.71 

29.21 

25.84 

29.10 

25.97 

39 

40 

30.19 

26.24 

30.07 

26.37 

29.96 

26.50 

29.84 

26  .  64 

40 

41 

30.94 

26.90 

30.83 

27.03 

30.71 

27.17 

30.59 

27.30 

41 

42 

31.70 

27.55 

31.58 

27.69 

31.46 

27.83 

31.33 

27.97 

42 

43 

32.45 

28.21 

32.33 

28.35 

32.21 

28.49 

32.08 

28.63 

43 

44 

33.21 

28.87 

33.08 

29.01 

32.95 

29.16 

32.83 

29.30 

44 

45 

33.96 

29.52 

33.83 

29.67 

33.70 

29.82 

33.57 

29.97 

45 

46 

34.72 

30.18 

34.58 

30.33 

34.45 

30.48 

34.32 

30.63 

46 

47 

35.47 

30.83 

35.34 

30.99 

35.20 

31.14 

35.06 

31.30 

47 

48 

36.23 

31.49 

36.09 

31.65 

35.95 

31.81 

35.81 

31.96 

48 

49 

36.98 

32.15 

36.84 

32.31 

36.70 

32.47 

36.56 

32.63 

49 

50 

37.74 

32.80 

37.59 

32.97 

37.45 

33.13 

37.30 

33.29 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

cJ 
o 

e. 

5 

49  Deg. 

48|  Deg. 

48£  Deg. 

484  Deg. 

n 

s 

TRAVERSE    TABLE. 


85 


5 

41  Deg. 

4H  Deg. 

41*  Deg. 

41|  Deg. 

s 

• 

s 

g. 

I 

Lat. 

Dep. 

Lat.   |   Dep, 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

38.49 

33.46 

38.34 

33.63 

38.20 

33.79 

38.05 

33.96 

51 

52 

39.24 

34.12 

39,10 

34.29 

38.95 

34.46 

38.79 

34.63 

S3 

53 

40.00 

34.77 

39.85 

34.95 

39.69 

35.12 

39.54 

35.29 

53 

54 

40.75 

35.43 

40.60 

35.60 

40.44 

35.78 

40.29 

35.96 

54 

55    41.51 

36.08 

41.35 

36.26 

41.19 

36.44 

41.03 

36.62 

55 

56 

42.26 

36.74 

42.10 

36.92 

41.94 

37.11 

41.78 

37.29 

56 

57 

13.02 

37.40 

42.85 

37.58 

42.69 

37.77 

42.53 

37.96 

57 

58 

43.77 

38.05 

43.61 

38.24 

43.44 

38.43 

43.27 

38.62 

58 

59 

44.53 

38.71 

44.36 

38.90 

44.19 

39.09 

44.02 

39.29 

59 

60 

45.28 

39.36 

45.11 

39.56 

44.94 

39.76 

44.76 

39.95 

60 

61 

46.04 

40.02 

45.86 

40.22 

45  .  69 

40.42 

45.51 

40.62 

61 

62 

4f,.79 

40.68 

46.61 

40.88 

46.44 

41.08 

46.26 

41.28 

62 

63 

47.55 

41.33 

47.37 

41.54 

47.18 

41.75 

47.00 

41.95 

63 

64 

48.30 

41.99 

48.12 

42.20 

47.93 

42.41 

47.75 

42.62 

64 

65 

49.06 

42.64 

48.87 

42.86 

48.68 

43.07 

48.49 

43.28 

65 

66 

49.81 

43.30 

49.62 

43.52 

49.43 

43.73 

49.24 

43.95 

66 

67 

50.57 

43.96 

50.37 

44.18 

50.18 

44.40 

49.99 

44.61 

67 

68 

51.32 

44.61 

51.13 

44.84 

50  .  93 

45.06 

50.73 

45.28 

68 

69 

52.07 

45.27 

51.88 

45.49 

51.68 

45.72 

51.48 

45.95 

69 

70 

52.83 

45.92 

52.63 

46.15 

52.43 

46.38 

52.22 

46.61 

70 

71 

53.58 

46.58 

53.38 

46.81 

53.18 

47.05 

52.97 

47.28 

71 

72 

54.34 

47.24 

54.13    47.47 

53.92 

47.71 

53.72 

47.94 

72 

73 

55.09 

47.89 

54.88  148.13 

54.67 

48.37 

54.46 

48.61 

73 

74 

55.85 

48.55 

55.64 

48.79 

55.42 

49.03 

55.21 

49.28 

74 

75 

56.60 

49.20 

56.39 

49.45 

56.17 

49.70 

55.95 

49.94 

75 

76 

57.36 

49.86 

57.14 

50.11 

56.92 

50.36 

56.70 

50.61 

76 

77 

58.1! 

50.52 

57.89 

50.77 

57.67 

51.02 

57.45 

51.27 

77 

78 

58.87 

51.17 

58.64 

51.43 

58.42 

51.68 

58.19 

51.94 

78 

79 

59.62 

51.83 

59.40 

52.09 

59.17 

52.35 

58.94 

52.60 

79 

80 

60.38 

52.48 

60.15 

52.75 

59.92 

53.01 

59.68 

53.27 

80 

81 

61.13 

53.14 

60.90 

53.41 

60.67 

53.67 

60.43 

53.94 

81 

82 

61.89 

53.80 

61.65 

54.07 

61.  4J 

54.33 

61.18 

54.60 

82 

83 

62.64 

54.45 

62.40 

54.73 

62.16 

55.00 

61.92 

55.27 

83 

84 

63.40 

55.11 

63.15 

55.38 

62.91 

55.66 

62.67 

55.93 

84 

85 

64.15 

55.76 

63.91 

56.04 

63.66 

56.32 

63.41 

56.60 

85 

86 

64.90 

56.42 

64.66 

56.70 

64.41 

56.99 

64.16 

57.27 

86 

87 

65.66 

57.08 

65.41 

57.36 

65.16 

57.65 

64.91 

57.93 

87 

88 

66.41 

57.73 

66.16 

58.02 

65.91 

58.31 

65.65 

58.60 

88 

89 

67.17 

58.39 

66.91 

58.68 

66.66 

58.97 

66.40 

59.26 

89 

90 

67.92 

59.05 

67.67 

59.34 

67.41 

59.64 

67.15 

59.93 

90 

91 

68.68|  59.70 

68.42 

60.00 

68.15 

60.30 

67.89 

60.60 

91 

92 

69.43 

60.36 

69.17 

6-0.66 

68.90 

60.96 

68.64 

61.26 

92 

93 

70.19 

61.01 

69.92 

61.32 

69.65 

61.62 

69.38 

61.93 

93 

94 

70.94 

61.67 

70.67 

61.98 

70.40 

62.29 

70.13 

62.59 

94 

95 

71.70 

62.33 

71.43 

62.64 

71.15 

62.95 

70.88 

63.26 

95 

96 

72.45 

62.98 

72.18 

63.30 

71.90 

63.61 

71.62 

63.92 

96 

97 

73.21 

63.64 

72.93 

63.96 

72.65 

64.27 

72.37 

64.59 

97 

98 

73.96 

64.29 

73.68 

64.62 

73.40 

64.94 

73.11 

65.26 

98 

99 

74.72 

64.95 

74.43 

65.28 

74.15 

65.60 

73.86 

60.  92 

99 

100 

75.47 

65.61 

75.18 

65.93 

74.90 

66.26 

74.61 

66.59 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

o 

V 

a 

I 

5 

3 

49  Deg. 

48|  Deg. 

48£  Deg. 

48*  Deg. 

P 

TBAVERSE    TABLE. 


d 

42  Deg. 

424  Deg. 

42i  Deg. 

42|  Deg. 

O 

B* 

P 

1 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

g 

1 

0.74 

0.67 

0.74 

0.67 

0.74 

0.68 

0.73 

0.68 

1 

2 

1.49 

1.34 

1.48 

1.34 

1.47 

1.35 

1.47 

1.36 

2 

3 

2.23 

2.01 

2.22 

2.02 

2.21 

2.03 

2.20 

2.04 

3 

4 

2.97 

2.68 

2.96 

2.69 

2.95 

2.70 

2.94 

2.72 

4 

5 

3.72 

3.35 

3.70 

3.36 

3.69 

3.38 

3.67 

3.39 

5 

6 

4.46 

4.01 

4.44 

4.03 

4.42 

4.05 

4.41 

4.07 

6 

7 

5.20 

4.68 

5.18 

4.71 

5.16 

4.73 

5.14 

4.75 

7 

8 

5.95 

5.35 

5.92 

5.38 

5.90 

5.40 

5.87 

5.43 

8 

9 

6.69 

6.02 

6.66 

6.05 

6.64 

6.08 

6.61 

6.11 

9 

10 

7.43 

6.69 

7.40 

6.72 

7.37 

6.76 

7.34 

6.79 

10 

11 

8.17 

7.36 

8.14 

7.40 

8.11 

7.43 

8.08 

7.47 

11 

12 

8.92 

8.03 

8.88 

8.07 

8.85 

8.11 

8.81 

8.15 

.  12 

13 

9.66 

8.70 

9.62 

8.74 

9.58 

8.78 

&.55 

8.82 

13 

14 

10.40 

9.37 

10.36 

9.41 

10.32 

9.46 

10.28 

9.50 

14 

15 

11.15 

10.04 

11.10 

10.09 

11.06 

10.13 

11.01 

10.18 

15 

16 

11.89 

10.71 

11.84 

10.76 

11.80 

10.81 

11.75 

10.86 

16 

17 

12.63 

11.38 

12.58 

11.43 

12.53 

11.48 

12.48 

11.54 

17 

18 

13.38 

12.04 

13.32 

12.10 

13.27 

12.16 

13.22 

12.22 

18 

19 

14.12 

12.71 

14.06 

12.77 

14.01 

12.84 

13.95 

12.90 

19 

20 

14.86 

13.38 

14.80 

13.45 

14.75 

13.51 

14.69 

13.58 

20 

21 

15.61 

14.05 

15.54 

14.12 

15.48 

14.19 

15.42 

14.25 

21 

22 

16.35 

14.72 

16.28 

14.79 

16.22 

14.86 

16.16 

14.93 

22 

23 

17.09 

15.39 

17.02 

15.46 

16.96 

15.54 

16.89 

15.61 

23 

24 

17.84 

16.06 

17.77 

16.14 

17.69 

16.21 

17.62 

16.29 

24 

25 

18.58 

16.73 

18.51 

16.81 

18.43 

16.89 

18.36 

16.97 

25 

26 

19.32 

17.40 

19.25 

17.48 

19.17 

17.57 

19.09 

17.65 

26 

27 

20.06 

18.07 

19.99 

18.15 

19.91 

18.24 

19.83 

18.33 

27 

28 

20.81 

18.74 

20.73 

18.83 

20.64 

18.92 

20  .  56 

19.01 

28 

29 

21.55 

19.40 

21.47 

19.50 

21.38 

19.59 

21.30 

19.69 

29 

30 

22.29 

20.07 

22.21 

20.17 

22.12 

20.27 

22.03 

20.36 

30 

31 

23.04 

20.74 

22.95 

20.84 

22.86 

20.94 

22.76 

21.04 

31 

32 

23.78 

21.41 

23.69 

21.52 

23.59 

21.62 

23.50 

21.72 

32 

33 

24.52 

22.08 

24.43 

22.19 

24.33 

22.29 

24.23 

22.40 

33 

34 

25  .  27 

22.75 

25.17 

22.86 

25.07 

22.97 

24.97 

23.08 

34 

35 

26.01 

23.42 

25.91 

23.53 

25.80 

23  .  65 

25.70 

23.76 

35 

36 

26.75 

24.09 

26  .  65 

24.21 

26.54 

24.32 

26.44 

24.44 

36 

37 

27  50 

24.76 

27.39 

24.88 

27  .  28 

25.00 

27.17 

25.12 

37 

38 

28.24 

25  .43 

28.13 

2o.55 

28.02 

25.67 

27.90 

25.79 

38 

39 

28.98 

26.10 

28.87 

26.22 

28.75 

26.35 

28  .  64 

26.47 

39 

40 

29.731  26.77 

29.61 

26.81* 

29.49 

27.02 

29.37 

27.15 

40 

41 

30.47 

27.43 

30.35 

27.57 

30  .  23 

27.70 

30.11 

27.83 

41 

31.21 

28.10 

31.09 

28.24 

30.97 

28.37 

30.84 

28  .  5  1 

42 

43 

31.96 

28.77 

31.83 

28.91 

31.70 

29.05 

31.58 

29.19 

43 

44 

32.70 

29.44 

32.57 

29.58 

32.44 

29.73 

32.31 

29.87 

44 

45 

33.44 

30.11 

33.31 

30.26 

33.18 

30.40 

33.04 

30.55 

45 

46 

34.18 

30.78 

34.05 

30.93 

33.91 

31.08 

33.78 

31.22 

46 

47 

34.93 

31.45 

34.79 

31.60 

34.65 

31.75 

34.51 

31.90 

47 

48 

35.67 

32.12 

35.53 

32:27 

35.39 

32.43 

35  .  25 

32.58 

48 

49 

36.41 

32.79 

36.27 

32  .  95 

36.13 

33.10 

35.98 

33  .  26 

49 

50 

37.16 

33.46 

37.01 

33.62 

36.86 

33.78 

36  .  72 

33.94 

50 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

c 

c 
c 

*> 

cd 
ce 

3 

48  Deg. 

47|  Deg. 

47  i  Deg. 

474  Deg. 

5 

TRAVERSE    TABLE. 


87 


D 

42  Deg. 

424  Deg. 

424  Deg. 

42}  Deg. 

q 

O^ 

» 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

51 

37.90 

34.13 

37.75 

34.29 

37.60 

34.46 

37.45 

34.62 

51 

52 

38.64 

34.79 

38.49 

34.96 

38.34 

35.13 

38.18 

35.30 

52 

53 

39.39 

35.46 

39.23 

35.64 

39.08 

35.81 

38.92 

35.98 

53 

54 

40.13 

36.13 

39.97 

36.31 

39.81 

36.48 

39.65 

36.66 

54 

55 

40.87 

36.80 

40.71 

36.98 

40.55 

37.16 

40.39 

37.33 

55 

56 

41.62 

37.47 

41.45 

37.65 

41.29 

37.83 

41.12 

38.01 

56 

57 

42.36 

38.14 

42.19 

38.32 

42.02 

38.51 

41.86 

38.69 

57 

58 

43.10 

38.81 

42.93 

39.00 

42.76 

39.18 

42.59 

39.37 

58 

59 

43.85 

39.48 

43.67 

39.67 

43.50 

39.86 

43.32 

40.05 

59 

60 

44.59 

40.15 

44.41 

40.34 

44.24 

40.54 

44.06 

40.73 

60 

61 

45.33 

40.82 

45.15 

41.01 

44.97 

41.21 

44.79 

41.41 

61 

62 

46.07 

41.49 

45.89 

41.69 

45.71 

41.89 

45.53 

42.09 

62 

63 

46.82 

42.16 

46.63 

42.36 

46.45 

42.56 

46.26 

42.76 

63 

64 

47.56 

42.82 

47.37 

43.03 

47.19 

43.24 

47.00 

43.44 

64 

65 

48.30 

43.49 

48.11 

43.70 

47.92 

43.91 

47.73 

44.12 

65 

66 

49.05 

44.16 

48.85 

44.38 

48.66 

44.59 

48.47 

44.80 

66 

67 

49.79 

44.83 

49.59 

45.05 

49.40 

45.26 

49.20 

45.48 

67 

68 

50.53 

45.50 

50.33 

45.72 

50.13 

45.94 

49.93 

46.16 

68 

69 

51.28 

46.17 

51.07 

46.39 

50.87 

46.62 

50.67 

46  ..84 

69 

70 

52.02 

46.84 

51.82 

47.07 

J51.61 

47.29    51.40 

47.52 

70 

71 

52.76 

47.51 

52.56 

47.74 

|52.35 

47.97 

52.14 

48.19 

71 

72 

53.51 

48.18 

53.30 

48.41 

53.08 

48.64 

52.87 

48.87 

72 

73 

54.25 

48.85 

54.04 

49.08 

53.82 

49.32 

53.61 

49.55 

73 

74 

54.99 

49.52 

54.78 

49.76 

54.56 

49.99 

54.34 

60.23 

74 

75 

55.74 

50.18 

55.52 

50.43 

55.30 

50.67 

55.07 

50.91 

75 

76 

56.48 

50.85 

56.26 

51.10 

56.03 

51.34 

55.81 

51.59 

76 

77 

57.22 

51.52 

57.00 

51.77 

56.77 

52.02 

56.54 

52.27 

77 

78 

57.97 

52.19 

57.74 

52.44 

57.51 

52.70 

157.28 

52.95 

78 

79 

58.71 

52.86 

58.48 

53.12 

58.24 

53.37 

58.01 

53.63 

79 

80 

59.45 

53.53 

59.22 

53.79 

58.98 

54.05 

58.75 

54.30 

80 

81 

00.19 

54.20 

59.96 

54.46 

59.72 

54.72 

i  59.48 

54.98 

81 

82 

60.94 

54.87 

60.70 

55.13 

60.46 

55.40 

60.21 

55.66 

82 

83 

61.68 

55.54 

61.44 

55.81 

61.19 

56.07 

60.95 

56.34 

83 

84 

62.42 

56.21 

62.18 

56.48 

61.93 

56.75 

161.68 

57.02 

84 

85 

63.17 

56.88 

62.92 

57.15 

62.67 

57.43 

62.42 

57.70 

85 

86 

63.91 

57.55 

63.66 

57.82 

63.41 

58.10 

163.15 

58.38 

86 

87 

64.65 

58.21 

64.40 

58.50 

64.14 

58.78 

63.89 

59.06 

87 

88 

65.40 

58.88 

65.14 

59.17 

64.88 

59.45 

164.62 

59.73 

88 

89 

66.14 

59.55 

65.88 

59.84 

65.62 

60.13 

65.35 

60.41 

89 

90 

66.88 

60.22 

66.62 

60.51 

66.35 

60.80 

J66.09 

61.09 

90 

91 

67.63 

60.89 

67.36 

61.19 

67.09 

61.48 

;66.82 

61.77 

91 

92 

68.37 

61.56 

6S.10 

61.86 

67.83 

62.15 

1  67.56 

62.45 

92 

93 

69.11 

62.23 

68.84 

62.53 

68.57 

62.83 

68.29 

63.13 

93 

94 

69.86 

62.90 

69.58 

63.20 

69.30 

63.51 

69.03 

63.  8J 

94 

95 

70.60 

63.57 

70.32 

63.87 

70.04 

64.18 

69.76 

64.49 

95 

96 

71.34 

64.24 

71.06 

64.55 

70.78 

64.86 

70.49 

65.16 

96 

97 

72.08 

64.91 

71.80 

65.22 

71.52 

65.53 

71.23 

65.84 

97 

98 

72.83 

65.57 

72.54 

65.89 

72.25 

66.21 

71.96 

66.52 

98 

99 

73.57 

66.24 

73.28 

66.56 

72.99 

66.88 

72.70 

67.20 

99 

100 

74.31 

66.91 

74.02 

67.24 

73.73 

67.56 

73.43 

67.88 

100 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

8 

H 

i 

00 

5 

48  Deg. 

47}  Deg. 

474  Deg. 

47*  Deg. 

.1 

TRAVERSE    TABLE. 


o 

43  Deg. 

434  Deg. 

43i  Deg. 

43J  Deg. 

C 

1' 

ft 

po 

1 

CO 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

p 

1 

0.73 

0.68 

0.73 

0.69 

0.73 

0.69 

0.72 

0.69 

1 

2 

1.46 

1.36 

1.46 

1.37 

1.45 

1.38 

1.44 

1.38 

2 

3 

2.19 

2.05 

2.19 

2.06 

2.18 

2.07 

2.17 

2.07 

3 

4 

2.93 

2.73 

2.91 

2.74 

2.90 

2.75 

2.89 

2.77 

4 

5 

3.66 

3.41 

3.64 

3.43 

3.63 

3.44 

3.61 

3.46 

5 

6 

4.39 

4.09 

4.37 

4.11 

4.35 

4.13 

4.33 

4.15 

6 

7 

6.12 

4.77 

5.10 

4.80 

5.08 

4.82 

5.06 

4.84 

7 

8 

5.85 

5.46 

5.83 

5.48 

5.80 

5.51 

5.78 

5.53 

8 

9 

6.58 

6.14 

6.56 

6.17 

6.53 

6.20 

6.50 

6.22 

9 

10 

7.31 

6.82 

7.28 

6.85 

7.25 

6.88 

7.22 

6  92 

10 

11 

8.04 

7.50 

8.01 

7.54 

7.98 

7.57 

7.95 

7.01 

11 

12 

8,78 

8.18 

8.74 

8.22 

8.70 

8.26 

8.67 

8.30 

12 

13 

9.51 

8.87 

9.47 

8.91 

9.43 

8.95 

9.39 

8.99 

13 

14 

10.24 

9.55 

10.20 

9.59 

10.16 

9.64 

10.11 

9.68 

14 

15 

10.97 

10.23 

10.93 

10.28 

10.88 

10.33 

10.84 

10.37 

15 

16 

11.70 

10.91 

11.65 

10.96 

11.61 

11.01 

11.56 

11.06 

16 

17 

12.43 

11.59 

12.38 

11.65 

12.33 

11.70 

12.28 

11.76 

17 

18 

13.16 

12.38- 

13.11 

12.33 

13.06 

12.39 

13.00 

12.45 

18 

19 

13.90 

12.96 

13.84 

13.02 

13.78 

13.08 

13.72 

13.14 

19 

20 

14.63 

13.64 

14.57 

13.70 

14.51 

J3.77 

14.45 

13.83 

20 

21 

15.36 

14.32 

15.30 

14.39 

15.23 

14.46 

15.17 

14.52 

21 

22 

16.09 

15.00 

13.02 

15.07 

15.96 

15.14 

15.89 

15.21 

22 

23 

16.82 

15.69 

16.75 

15.76 

16.68 

15.83 

16.61 

15.90 

23 

24 

17.55 

16.37 

17.48 

16.44 

17.41 

16.52 

17.34 

16.60 

24 

2f> 

18.28 

17.05 

18.21 

17.13 

18.13 

17.21 

18.06 

17.29 

25 

26 

19.02 

17.73 

18.94 

17.81 

18.86 

17.90 

18.78 

17.98 

26 

27 

19.75 

18.41 

19.67 

18  50 

19.59 

18.59 

19.50 

18.67 

27 

28 

20.48 

19.10 

20.  3y 

19.19 

20.31 

19.27 

20.23 

19.36 

28 

29 

21.21 

19.  78 

21.12 

19.87 

21.04 

19.96 

20.95 

20.05 

29 

30 

21.94 

20.46 

21.85 

20.56 

21.76 

20.65 

21.67 

20.75 

30 

31 

22.67 

21.14 

22.58 

21.24 

22.49 

21.34 

22.39 

21.44 

31 

32 

23.40 

21.82 

23.31 

21.  9a 

23.21 

22.03 

23.12 

22.13 

32 

33 

24.13 

22.51 

24.04 

22.61 

23.94 

22.72 

23.84 

22.82 

33 

34 

24.87 

23.19 

24.76 

23.30 

24.66 

23.40 

24.56 

23.51 

34 

35 

25.60 

23.87 

25.49 

23.98 

25.39 

24.09 

25  .  28 

24.20 

35 

36 

26.33 

24.55 

26.22 

24  ..67 

26.11 

24.78 

26.01 

24.89 

36 

37 

27.06 

25.23 

26.95 

25.35 

26.84 

25.47 

26.73 

25.  5a 

37 

38 

27.79 

25.92 

27.68 

26.04 

27.56 

26.16 

27.45 

26.28 

38 

39 

28.52 

26.60 

28.41 

26.72 

28.29 

26.86 

28.17 

26.97 

39 

40 

29.25 

27.28 

29.13 

27.41 

29.01 

27.53 

28.89 

27.66 

40 

41 

29.99 

27.96 

29.86 

28.09 

29.74 

28.22 

29  .  62 

28.35 

"41 

42 

30.72 

28.64 

30.59 

28.78 

30.47 

28.91 

30.34 

29.04 

42 

43 

31.45 

29.33 

31.32 

29.46 

31.19 

29.60 

31.06 

29  .  74 

43 

44 

32.18 

30.01 

32.05 

30.15 

31.92 

30.29 

31.78 

30.43     44 

45 

32.91 

30.69 

32.78 

30.83 

32.64 

30.98 

32.51 

31.12 

45 

46 

33.64 

31.37 

33.51 

31.52 

33.37 

31.66 

33.23 

31.81 

46 

47 

34.37 

32.05 

34.23 

32.20 

34.09 

32.  35 

33.95 

32  .  50 

47 

48 

35.10 

32.74 

34.96 

32.89 

34.82 

33.04 

34.67 

33.19 

48 

49 

35.84 

33.42 

35.69 

33.57 

35.54 

33.73 

35.40 

33.88 

49 

50 

36  .  57 

34.10 

36.42 

34.26 

36.27 

34.42 

36.12 

34.58 

50 

§ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.. 

Lat. 

1 

1 

47  Deg. 

46|  Deg. 

46J  I>*g- 

46i  Deg. 

I 

b 

TB AVERSE    TABLE. 


89 


D 

43  Deg. 

43|  Deg. 

43£  Deg. 

43|  Deg. 

O 

P 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 
«     , 

51 

37.30 

34.78 

37.15 

34.94 

36.99 

35.11 

36.84 

35.27 

51 

52 

38.03 

35.46 

37.88 

35.63 

,37.72 

35.79 

37.56 

35.96 

52 

53 

38.76 

36.15 

38.60 

36.31 

38.44 

36.48 

38.29 

36.65 

53 

•54 

39.49 

36.83 

39.33 

37.00 

39.17 

37.17 

39.01 

37.34 

54 

55 

40.22 

37.51 

40.06 

37.69 

39.90 

37.86 

39.73 

38.03 

55 

56 

40.96 

38.19 

40.79 

38.37 

40.62 

38.55 

40.45 

38.72 

56 

57 

41.69 

38.87 

41.52 

39.06 

41.35 

39.24 

41.17 

39.42 

57 

58 

42.42 

39.56 

42.25 

39.74 

42.07 

39.92 

41.90 

40.11 

58 

59 

43.15 

40.24 

42.97 

40.43 

42.80 

40.61 

42.62 

40.80 

59 

60 

43.88 

40.92 

43.70 

41.11 

43.52 

41.30 

43.34 

41.49 

60 

61 

44.61 

41.60 

44.43 

41.80 

44.25 

41.99 

44.06 

42.18 

61 

62 

45.34 

42.28 

45.16 

42.48 

44.97 

42.68 

44.79 

42.87 

62 

63 

46.08 

42.97 

45.89  |43.17 

45.70 

43.37 

45.51 

43.57 

63 

64 

46.81 

43.65 

46.62 

43.85 

46.42 

44.05 

46.23 

44.26 

64 

65 

47.54 

44.33 

47.34 

44.54 

47.15 

44.74 

46.95 

44.95 

65 

66 

48.27 

45.01 

43.07 

45.22 

47.87 

45.43 

47.68 

45.64 

66 

67 

49.00 

45.69 

48.80 

45.91 

48.60 

46.1:* 

48.40 

46.33 

67 

68 

49.73 

46.38 

49.53 

46.59 

49.33 

46.81 

49.12 

47.02 

68 

69 

50.46 

47.06 

50.26 

47.28 

50.05 

47.50 

49.84 

47.71 

69 

70 

51.19 

47.74 

50.99 

47.96 

50.78 

48.18 

50.57 

48.41 

70 

71 

51.93 

48.42 

51.71 

4S.65 

51.50 

48.87 

51.29 

49.10 

71 

72 

52.66 

49.10 

52.44 

49.33 

52  .  23 

49.56 

52.01 

49.79 

72 

73 

53.39 

49.79 

53.17 

50.02 

52  .  95 

50.25 

52.73 

50.48 

73 

74 

54.12 

50.47 

53  .  90 

50.70 

53.68 

50.94 

53.45 

51.17 

74 

75 

54.85 

51.15 

54.63 

51.39 

54.40 

51.63 

54.18 

51.86 

75 

76 

55  .  58 

51.83 

55.36 

52.07 

55.13 

52.31 

54.90 

52.55 

76 

77 

56.31 

52.51 

56.08 

52.76 

55.85 

53.00 

55.62 

53.25 

77 

78 

57.05 

53.20 

56.81 

53.44 

56.58 

53.69 

56.34 

53.94 

78 

79 

57.78 

53.88 

57.54 

54.13 

57.30 

54.38 

57.07 

54.63 

79 

80 

58.51 

54.56 

58.27 

54.81 

58.03 

55.07 

57.79 

55.32 

80 

81 

59.24 

55.24 

59.00 

55.50 

58.76 

55.76 

58.51 

56.01 

81 

82 

59.97 

55.92 

59.73 

56.18 

59.48 

56.45 

59.23 

56.70 

82 

83 

60.70 

56.61 

60.45 

56.87 

60.21 

57.13 

59.96 

57.40 

83 

84 

61.43 

57.29 

61.18 

57.56 

60.93 

57.82 

60.68 

58.09 

84 

85 

62.17 

57.97 

61.91 

58.24 

61.66 

58.51 

61.40 

58.76 

85 

86 

62.90 

58.65 

62.64 

58  .  93 

62.38 

59.20 

62.12 

59.47 

86 

87 

63.63 

59.33 

63.37 

59.61 

63.11 

59.89 

62.85 

60.16 

87 

88 

64.36 

60.02 

64.10 

60.30 

63.83 

60.58 

63.57 

60.85 

88 

89 

65.09 

60.70 

64.82 

60.98 

64.56 

61.26 

64.29 

61.54 

89 

90 

65.82 

61.38 

65.55 

61.67 

65.28 

61.95 

65.01 

62,24 

90 

91 

66.55 

62.06 

66.28 

62.35 

66.01 

62.64 

65.74 

62.93 

91 

92 

67.28 

62.74 

67.01 

60  .  04 

66.73 

63.33 

66.46 

63.62 

92 

93 

68.02 

63.43 

67.74 

63.72 

67.46 

64.02 

67.18 

64.31 

93 

94 

68.75 

64.11 

68.47 

64.41 

68.19 

64.71 

67.90 

65.00 

94 

95 

69.48 

64.79 

69.20 

65.09 

68.91 

65.39 

68.62 

65.69 

95 

96 

70.21 

65.47 

69.92 

65.78 

69.64 

66.08 

69.35 

66.39 

96 

97    70.94 

66.15 

70.65 

66.46 

70.36 

66.77 

70.07 

67.08 

97 

98 

71.67 

66.84 

71.37 

67.15 

71.09 

67.46 

70.79 

67.77 

98 

99 

72.40 

67.52 

72.11 

67.83 

71.81 

68.15 

71.51 

68.46 

99 

100 

73.14 

68.20 

72.84 

68.52 

72.54 

68.84 

72.24 

69.15 

100 

8 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

c. 

e 
c 

1 

a 

cc 

$ 

47  Deg. 

46|  Deg. 

46£  Deg. 

46$  Deg. 

5 

TRAVERSE    TABLE. 


g 

ST 

44  Deg. 

444  Deg. 

44£  Deg. 

44|  Deg. 

45  Deg. 

G 
B' 

P 

1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 

8 

1 

0.72 

0.69 

0.73 

0.70 

0.71 

0.70 

0.71 

0.71 

0.71 

~OT71 

i 

2 

1.44 

1.39 

1.43 

1.40 

1.43 

1.40 

1.42 

1.41 

1.41 

1.41 

2 

3 

2.16 

2.08 

2.15 

2.09 

2.14 

2.10 

2.13 

2.11 

2.12 

2.12 

3 

4 

2.88 

2.78 

2.87 

2.79 

2.85 

2.80 

2.84 

2.82 

2.83 

2.83 

4 

5 

3.60 

3.47 

3.58 

3.49 

3.5? 

3.50 

3.55 

3.52 

3.54 

3.54 

5 

6 

4.32 

4.17 

4.30 

4.19 

4.28 

4.21 

4.26 

4.22 

4.24 

4.24 

6 

7 

5.04 

4.86 

5.01 

4.88 

4.99 

4.91 

4.97 

4.93 

4.95 

4.95 

7 

8 

5.75 

5.56 

5.73 

5.58 

5.71 

5.61 

5.68 

5.63 

5.66 

5.66 

8 

9 

6.47 

6.25 

6.45 

6.28 

6.42 

6.31 

6.39 

6.34 

6.36 

6.36 

9 

10 

7.19 

6.95 

7.16 

6.98 

7.13 

7.01 

7.10 

7.04 

7.07 

7.07 

10 

11 

7.91 

7.64 

7.88 

7.68 

7.85 

7.71 

7.81 

7.74 

7.78 

7.78 

11 

12 

8.63 

8.34 

8.60 

8.37 

8.56 

8.41 

8.52 

8.45 

8.49 

8.49 

12 

13 

9.35 

9.03 

9.31 

9.07 

9.27 

9.11 

9.23 

9.15 

9.19 

9.19 

13 

14 

10.07 

9.73 

10.03 

9.77 

9.99 

9.81 

9.94 

9.86 

9.90 

9.90 

14 

15 

10.79 

10.42 

10.74 

10.47 

10.70 

10.51 

10.65 

10.56 

10.61 

10.61 

15 

16 

11.51 

11.11 

11.46 

11.16 

11.41 

11.21 

11.36 

11.26 

11.31 

11.31 

16 

17 

12.23 

11.81 

12.18 

11.86 

12.13 

11.92 

12.07 

11.97 

12.02 

12.02 

17 

18 

12.95 

12.50 

12.89 

12.56 

12.84 

12.62 

12.78 

12.67 

12.73 

12.73 

18 

19 

13.67 

13.20 

13.61 

13.26 

13.55 

13.32 

13.49 

13.38 

13.43 

13.43 

19 

20 

14.39 

13.89 

14.33 

13.96 

14.26 

14.02 

14.20 

14.08 

14.14 

14.14 

20 

21 

15.11 

14.59 

15.04 

14.65 

14.98 

14.72 

14.91 

14.78 

14.85 

14.85 

21 

22 

15.83 

15.28 

15.76 

15.35 

15.69 

15.42 

15.62 

15.49 

15.56 

15.56 

22 

23 

16.54 

15.98 

16.47 

16.05 

16.40 

16.12 

16.33 

16.19 

16.26 

16.26 

23 

24 

17.26 

16.67 

17.19 

16.75 

17.12 

16.82 

17.04 

16.90 

16.97 

16.97 

24 

25 

17.98 

17.37 

17.91 

17.44 

17.83 

17.52 

17.75 

17.60 

17.68 

17.68 

25 

26 

18.70 

18.06 

18.62 

18.14 

18.54 

18.22 

18.46 

18.30 

18.38 

18.38 

26 

27 

19.42 

18.76 

19.34 

18.84 

19.26 

18.92 

19.17 

19.01 

19.09 

19.09 

27 

28 

20.14 

19.45 

20.06 

19.54 

19.97 

19.63 

19.89 

19.71 

19.80 

19.80 

28 

29 

20.86 

20.15 

20.77 

20.24 

20.68 

20.33 

20.60 

20.42 

20.51 

20.51 

29 

30 

21.58 

20.84 

21.49 

20.93 

21.40 

21.03 

21.31 

21.12 

21.21 

21.21 

30 

31  22.30 

21.53 

22.21 

21.63 

22.11 

21.73 

22.02 

21.82 

21.92 

21.92S31 

32 

23.02 

22.23 

22.92 

22.33 

22.82 

22.43 

22.73 

22.53 

22.63 

22.63 

32 

33 

23.74 

22.92 

23.64 

23.03 

23.54 

23.13 

23.44 

23.23 

23.33 

23.33 

33 

34 

24.46 

23.62 

24.35 

23.72 

24.25 

23.83 

24.15 

23.94 

24.04 

24.04 

34 

3525.18 

24.31 

25.07 

24.42 

24.96 

24.53 

24.86 

24.64 

24.75 

24.7535 

36125.90 

25.01 

25.79 

25.12 

25  .  68 

25  .  23 

|25.57 

25.34 

25.46 

25.46136 

37 

26.62 

25.70 

26.50 

25.82 

26.39 

25.93 

26.28 

26.05 

26.16 

26.16 

37 

38 

27.33 

26.40 

27.22 

26.52 

27.10 

26.63 

|26.99 

26.75 

26.87 

26.87 

38 

39 

28.05 

27.09 

27.94 

27.21 

27.82 

27.34 

,27.70 

27.46 

27.58 

27.58 

39 

40 

28.77 

27.79 

28.65 

27.91 

28  .  53 

28.04 

,28.41 

28.16 

28.28 

28.28 

40 

41 

29.49 

28.48 

29.37 

28.61 

29.24 

28.74 

29.12 

28.86 

28.99 

28.99 

41 

42 

30.21 

29.18 

30.08 

29.31 

29.96 

29.44 

29.83 

29.57 

29.70 

29.70 

42 

43 

30.93 

29.87 

30.80 

30.00 

30.67 

30.14 

30.54 

30.27 

30.41 

30.41 

43 

44 

31.65 

30.56 

31.52 

30.70 

31.38 

30.84 

31.25 

30.98 

31.11 

31.11 

44 

45 

32.37 

31.26 

32.23 

31.40 

32.10 

31.54 

31.96 

31.68 

31.82 

31.82 

45 

46 

33.09 

31.95 

32.95 

32.10 

32.81 

32.24 

32.67 

32.38 

32.53 

32.5346 

47 

33.81 

32.65 

33.67 

32.80 

33.52 

32.94 

33.38 

33.09 

33.23 

33.23 

47 

48 

34.53 

33.34 

34.38 

33.49 

34.24 

33.64 

34.09 

33.79 

33.94 

33.94 

48 

49 

35  .  25 

34.04 

35.10 

34.19 

34.95 

34.34 

34.80 

34.50 

34.65 

34.65 

49 

50 

35.97 

34.73 

35.82 

34.89 

35.66 

35.05 

35.51 

35.20 

35.36 

35.36 

50 

I 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

OB 

Q 

46  Deg. 

45|  Deg. 

45£  Deg. 

45|  Deg. 

45  Deg. 

d 

"02 

Q 

TRAVERSE   TABLE. 


9i 


y      44  Deg. 
ST 

444  Deg. 

44i  Deg. 

44|  Deg. 

45  Deg. 

D 

1' 

§    Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

3 
o 

CD 

51  36.69 

35.43 

36.53 

35.59 

36.3835.75 

36.22 

35.90:36.06 

36.06 

51 

52  37.41 

36.12 

37.25 

36.29 

37.09  36.45 

36.93 

36.61  36.77 

36.77 

52 

5333.12 

36.82 

37.96 

36.98 

37.8037.15 

37.64 

37.31  37.48 

37.48 

53 

54  38.84 

37.51  38.68 

37.68 

38.5237.85 

38.35 

38.02 

38.18 

38.18 

54 

5539.56 

38.21 

39.40 

38.38 

39.2338.55 

39.06 

38.72 

38.89 

38.89 

55 

56  40  .  28 

38.90 

40.11 

39.08 

39.94'39.25 

39.77 

39.42 

39.60 

39.60 

56 

5741.0039.60 

40.83 

39.77 

40.6639.95 

40.48 

40.13 

40.31 

40.31 

57 

58,11.72 

40.29 

41.55 

40.47 

41.3740.65 

41.19 

40.83 

41.01 

41.01 

58 

5942.44 

40.98 

42.26 

41.17 

42.0841.35 

41.90 

41.54 

41.72 

41.72 

59 

6043.16 

41.68 

42.98 

41.87 

42.7942.05 

42.61 

42.24 

42.43 

42.43 

60 

61  43.8842.37 

43.69 

42.57 

43  51  42.76 

43.32 

42.94J43.13 

43.13 

61 

62  44.6043.07 

44.41 

43.26 

44  22 

43.46 

44.03 

43.65 

43.84 

43.84 

62 

6345.32J43.76 

45.13 

43.96 

44  9344.16 

44.74 

44.35 

44.55 

44.55 

63 

6446.04 

44.46 

45.84 

44.66 

45  6544.86 

45.45 

45.06 

45.25 

45.25 

64 

65146.  76!45.  15 

46.56 

45.36 

46.3645.56 

46.16 

45.76 

45.96 

45.96 

65 

66  47.48 

45.85 

47.28 

46.05 

47.07i46.26 

46.87 

46.46 

46.67 

46.67 

66 

8748.20 

46.54 

47.9946.75 

47.79 

46.96 

47.58 

47.17 

47.38 

47.38 

67 

6848.9? 

47.24 

48.71  47.45  48.50 

47.66 

48.29 

47.87 

48.08 

48.08 

68 

6949.6347.93 

49.  42148.  15  49.21 

48.36 

49.00 

48.58 

48.79 

48.79 

69 

70J50.35 

48.63 

50.  H  48.85'  49.  93 

49.06 

49.71 

49.28 

49.50 

49.50 

70 

71i51.0749.32 

50.86J49.54l50.64 

49.761,50.42 

49.98 

50.20150.20 

71 

72  51.79 

50.02 

51.57 

50.34851.85 

50.  47  151.13  50.  69 

50.91  50.91 

72 

7352.51 

50.71 

52.29 

50.94  52.07 

51.17 

51.8451.39 

51.  62151.  62 

73 

7453.23 

51.40 

53.01 

51.64J52.78 

51.87!,I52.55!52.10 

52.33J52.33 

74 

7553.95 

52.10 

53.72 

52.33j|53.49 

52.57i53.26|52.80 

53.0353.03 

75 

7654.67 

52.79 

54.44 

53.03  54.21 

53.  27)53.  97  53.  51 

53.7453.74 

76 

77|55.39  53.49 

55.16 

53.  73.54.92 

53.97  54.6854.21 

54.45154.45 

77 

78)56.11 

54.18 

55.87 

54.43J 

55.63 

54.67155.3954.91 

55.1555.15 

78 

79  56.83 

54.88 

56.59 

55.13 

56.35 

55.37 

56.10  55.62 

55.86 

55.86 

79 

8057.55 

55.57 

57  .  30 

55.82 

57.06 

56.07 

56.81 

56.32 

56.5756.57 

80 

81  58.27 

56.27 

58.02 

56.52 

57.77 

56  .  77 

57.5257.03 

57.2857.28    81 

8258.99 

56.96 

58.74 

57.22 

58.49 

57.47 

58.  24|57.73 

57.9857.98 

82 

8359.71 

57.66 

59.45 

57.92 

59.20 

58.18 

58.9558.43 

58.6958.69 

83 

8160.42 

58.35 

60.17 

58.61 

59.91 

58.88 

59.66 

59.14 

59.4059.40 

84 

8561.14 

59.05 

60.89 

59.31 

60.63 

59.58 

60.37 

59.84 

60.1060.10 

85 

86'61.86 

59  .  74 

61.60 

60.01 

61.34 

60.28 

61.0860.55 

60.  81160.  81 

86 

87162.53 

60.44 

62.32 

60.71 

62.05 

60.98 

61.79:61.25 

61.5261.52 

87 

88163.30 

61.13 

63.03 

61.41 

62.77 

61.68 

62.50i61.95 

62.2362.23 

88 

89:64.0261.82 

63.7562.10 

63.48 

62.38 

63.21  62.66  62.93  62.93 

89 

90.64.7462.52 

64.4762.80 

64.19 

63.08 

63.92  63.  36:;63.  64:63.  64    90 

9L65.46 

63.21 

65.18 

63.50  64.91 

63.78 

64.63  64.07  64.35  64.35    91 

92  66.18 

63.91 

65.90 

64.20  65.62 

64.48 

65.34  64.77 

65.0565.05    92 

93  66.90 

64.60 

66.62 

64.89  66.33 

65.18 

66.0565.47 

65.76  65.76 

93 

94:67.62 

65.30 

67.33 

65.59  67.05 

65.89 

66.7666.1866.47:66.47    94 

9o  68.  34  65.  99  63.0566.29! 

67.76 

66.59 

67.  47)66.  88:67.  18i67.  18 

95 

9ri  69.0666.69 

68.76 

66.99 

68.47 

67.29  68.  18167.59  |67.88  67.  88 

96 

97;69.78 

67.38 

69.48 

67.69 

69.19 

67.99  168.89  68.29,  68.59  68.59 

97 

93  70.50 

68.08 

70.20 

68.38 

69.90 

68.69  69.60 

68.99  69.30  69.30 

98 

99  71.21 

68.77 

70.91 

69.08 

70.61 

69.39 

70.31 

69.70 

70.00 

70.00 

99 

100  71.93 

69.47 

71.63 

69.78 

71.33 

70.09 

71.02 

70.40 

70.71 

70.71 

100 

|     Dep. 

Lat. 

Dep.    Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

» 

6 

c 

7> 

1 

2        46  Deg. 

45?  Deg. 

45  i  Deg. 

45*  Deg. 

45  Deg. 

~ 

A  TABLE  OF  NATURAL  SIXES. 


0  JDeg. 

1  Deg. 

2  L)eg. 

3  Deg. 

4  Deg. 

.Nut. 

N.  Co- 

"ssn 

V.  Co- 

Nat. 

V  Co- 

Nat.  ] 

V.  Co- 

Nat. 

V.  Co- 

M 

Sine 

Sii.e 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

M 

0 

00000 

Unil. 

01745J99985 

03490 

9939 

05234 

)9863 

06976 

J9756  t 

iO 

1 

00029 

00000 

01774 

99984 

03519 

9938 

05263:99861 

07005 

59754  i 

>9 

2 
3 

00058 
00087 

00000 
00000 

01803 
01832 

99984 
99983 

03548 
03577 

9937 
9936 

05292  99860 
0532ll99858 

07034 
07063 

^9752  { 
99750  1 

>8 
37 

4 

00116 

00000 

01862 

99983 

03606 

99935 

05350*99857 

07092 

99748  . 

36 

5 

0014.0 

00000 

01891 

99982 

03635 

99934 

05379  99855 

07121 

99746  , 

35 

6 

00175 

00000 

01920 

99982 

03664 

99933 

05408199854 

07150 

99744 

34 

7 

00204 

00000 

01949 

99981 

03693 

99932 

05437 

99852 

07179 

99742 

33 

8 

00233 

00000 

01978 

99980 

03723 

99931 

05466 

99851 

07208 

99740 

32 

9 

00262 

00000 

02007 

99980 

03752 

99930 

05495 

99849 

07237 

99738 

31 

10 

00291 

00000 

02036 

99979, 

03781 

09929 

05524 

99847 

07266 

99736 

0 

11 

0032099999 

02065 

99979 

03810 

99927 

05553^99846 

07295 

99734 

19 

12 

00349  99999 

02094 

99978 

03839 

99926 

05582 

99844 

07324 

99731 

48 

13 

00378 

99999 

02123 

99977 

03868 

99925 

05611 

99842 

07353 

99729 

17 

14 

00407 

99999 

02152 

99977 

03897 

49924 

05640 

99841 

07382 

99727 

46 

15  Gu436 

99999 

02181 

99976 

03926 

J9923 

05669 

99839 

07411 

99725 

45 

16 

00465 

9999  ' 

022  1  1 

99976 

03955 

99922 

05698 

99838 

07440 

99723 

44 

17 

00495 

99999 

02240 

99975 

03984 

99921 

05727 

99836 

07469 

99721 

43 

18J00524 

99999 

02269 

99974 

04013 

J9919 

05756 

99834 

07498 

99719 

42 

19  00553 

99998 

02298 

99974 

04042 

99918 

05785 

99833 

07527 

99716 

41 

20 

00582 

99998 

02327 

99973 

04071 

99917 

05814 

99831 

07556 

99714 

40 

21 

006  1  1 

99998 

02356 

99972 

04100 

99916 

05844 

99829 

07585 

99712 

39 

22 

00640 

99998 

02385 

99972 

04129 

9991* 

05873 

99827 

07614 

99710 

38 

23 

00669 

99998 

02414 

99971 

04159 

99913 

05902 

99826 

07643 

99708 

37 

24 

00698  99998 

02443 

99970 

04188 

99912 

05931 

99824 

07672 

99705 

36 

2500727 

99997 

02472 

99969 

04217 

J9911 

05960 

99822 

07701 

99703 

35 

26t007f'6 

99997 

02501 

99969 

04246 

99910 

05989  9982 

07730 

99701 

34 

27(00785 

99997 

02530 

99968 

04275 

99909 

06018 

99819 

07759 

99699 

33 

28i008l4!99997 

02560 

9996*7 

04304 

99907 

06047 

99817 

07788 

99696 

32 

29 

00844 

99996 

02589 

99966 

04333 

99906 

06076 

99815 

07817 

99694 

31 

30 

00873 

99996 

02618 

99966 

04362 

J9905 

06105 

99813! 

07846 

99692 

30 

3L00902 

99996 

02647 

99965|  04391 

99L04 

06134 

998121 

07875 

99689 

29 

32!  00931 

99996 

02676 

099  64 

04420 

99902 

06163 

99810 

07904 

99687 

28 

33100960 

99995 

02705 

99963 

04449 

99901 

06192 

99808- 

07933 

99685 

27 

34100989 

99995 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35  !0  JO  18  999.95 

02763 

99962 

04507 

99898 

06250 

99804 

07991 

99680 

25 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06279 

99803 

08020 

99678 

2* 

37 

01076 

99994 

32821 

99960 

04565 

99896 

06308 

99801 

08049 

99676 

23 

38 

01105199994 

02-!  50 

99959 

04594 

99894 

06337 

99799 

08078 

99673 

22 

39 

01131  99994 

02879 

99959 

04623 

99893 

06366 

99797 

08107 

99671 

21 

40 

01164 

99993 

02908 

09958  04653 

99892 

06395 

99795 

08136 

99668 

20 

41  01193 

42101222 

99993 
99993 

02938 
02967 

99957  04682 
99956:04711 

99890 
99889 

06424 
06453 

99793 
99792 

.08165 
08194 

99666 
99664 

19 

18 

43:  01251 

99992 

02996 

99»65  04740 

99888 

06482 

99790 

08223 

99661 

17 

44101280 

99992 

03025 

99954  104769 

99886 

06511 

99788 

08252 

99659 

16 

45  01309 

99991 

03054 

99953.0479S 

99885 

06540 

99786 

08281 

99657 

15 

46iO  1338|99991 

03083 

09952  ':  04827 

99883 

06569 

99784 

08310 

99654 

14 

4701367 

99991 

03112 

09952  04856 

99882 

06598 

99782 

08339 

99652 

13 

48iOl39« 

99990 

03141 

99951:04885 

99881 

06627 

99780 

08368 

99649 

12 

49 

01425 

99990 

03170 

99950  04914 

99879 

06656 

99778 

08397  99647 

11 

50 

01454 

99989 

03199 

99949  S04943  99878 

06685 

99776 

08426  99644 

1 

51 

01483 

99989 

03228 

99948  04972!99876 

06714 

99774 

08455  99642 

52 

01513 

99989 

03257 

99947  05001  99875 

06743 

99772 

08484  99639 

53 

01542 

99988 

03286 

99946  05030  99873 

06773.99770 

08513  99637 

54 

01571 

99988 

03316 

99945  05059  99872  i|06802  .99768 

08542  99635 

55 

01600 

99987 

03345 

99944  05088  99870 

06831 

99766 

08571  99632 

5f 

01629 

99987 

03374 

99943  0511799869 

06860 

99764 

08600  99630 

I 

5701658 

99986 

03403 

99942  05146 

99867 

06889 

99762 

9862S 

99627 

5801687 

99986 

03432 

99941  0517f 

99866 

06918 

99760 

0865? 

99625 

*  $ 

59J01716 

99985 

03461 

99940  0520£ 

99864 

06947 

99758 

08687 

99622 

M 

N.  Co- 

Nat. 

N.  Co- 

Nat. 

N.  Co- 

Nat. 

N.  Co- 

Nat. 

\.  Co- 

Nat. 

M 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

89  Deff. 

88  He?. 

87  Dee. 

86  Dejr. 

85  Deg. 

A   TABLE   OF   WATTTRAl   S1WES. 


o  L)eg. 

7  Deg. 

I   8  Deg. 

9  Deg. 

M 

N.  8. 

N.  CS. 

N.S. 

N.CS. 

N.S. 

N.  CS. 

N.S. 

N.CS. 

N.S. 

N.C8. 

M 

0 

08716 

99619 

10453 

99452 

12187 

99255 

13917 

99027; 

15643 

98769 

"0 

1 

08745 

99617 

10482 

99449 

12216 

99251 

13946 

99023 

15672 

98764 

9 

2 

08774 

99614 

10511 

99446 

12245 

99248 

13975 

9P019: 

15701 

98760 

8 

3 

0880399612 

10540 

99443 

12274 

99244J:i4004!990i5! 

15730 

98755 

7 

4 
5 

08831  '99609  1056999440 
0886(>!99607l  10597  99437 

12302 
12331 

99240 
99237 

1403399011 
14061  99006 

15758 
15787 

98751 
98746 

6 
5 

6 

0888999604 

10626 

99434 

12360199233 

14090  99002i 

15816 

98741 

4 

7  08918'99602 
8(0894799599 

10655 
10684 

99431 
99428 

12389  99230 
12418199226 

141i9 
14148 

98998 
98994 

15845 
15873 

98737 
98732 

3 

2 

910897699596 

10713 

99424 

12447199222 

14177 

98990 

15902 

98728 

1 

10 

09005  99594 

10742 

99421 

12476 

99219 

14205  98986 

15931 

98723 

0 

11 

09034  99591 

10771 

99418 

12504 

99215 

,14234 

9S9S2 

15959 

98718 

9 

12 

09063  99588 

10800 

99415 

12533 

99211 

14263  98978 

15988 

98714 

48 

13  09092  99586 

10829 

99412 

12562 

99208 

14292 

98973 

16017 

98709 

47 

1409121  99533 

10858 

99409 

12591 

99204 

14320 

98969 

16046 

98704 

46 

15 

09150  99580 

10887 

99406 

12620 

99200 

14349 

98965 

16074 

98700 

45 

16 

09179 

99578 

10916 

99402 

12649 

99197 

14378 

98961 

16103 

98695 

44 

17 

09208  99575 

10945 

99399 

12678 

99193 

14407 

98957 

16132 

98690 

43 

18  09237'  99572 

10973 

99396 

12706 

99189114436 

98953 

16160 

9*6*6 

42 

19  09266199570 
20!09295I99567 

11002 
11031 

99393 
99390 

12735 
12764 

99186 
99182 

J14464  98948  16189  98681 
14493  989441!  162  IS  28676 

41 
40 

21  09324i99564 

'  11060 

99386 

12793 

99178 

14522  98940  16246 

98671 

39 

22|09353!99562 

11089 

99383 

1282^ 

99175 

14551 

98936  16275 

98667 

38 

23  '09382!  99559 

11118 

99380 

12851 

99171 

14580 

98931 

16304 

98662 

37 

24'0941  1)99556 
25  09440:99553 

ill  147  99377 
11176199374 

12880 
12908 

99167 
99163 

14608 
14637 

98927 
98923 

16333  98657 
16361198652 

36 
35 

26 

09469  99551  ,11205 

99370 

12937 

99160!  14666J98919 

16390 

98648 

34 

27  09498  J99548 

11234 

99367 

12966 

99156  14695198914 

16419 

9SR43 

33 

28 

09527  99545:11263 

99364 

12995 

99152 

1472398910 

16447198638 

32 

29!09556l99542  111291 

99360 

13024 

99148 

1475298906 

16476  98633 

31 

30 

09585  99540,  11320  99357 

13053 

99144 

14781  98902 

16505198629 

30 

31 

09614 

99537  1134999354 

13081 

99141 

1481098897 

16533 

9?*  6  24 

29 

32 

09642 

99534  11378 

99351 

13110 

99137 

'14838 

98893 

16562 

98619 

28 

33 

0967] 

99531  11407 

99347 

13139 

99133 

114867 

98889 

16591 

'98614 

27 

34 

09700 

995281  1143699344 

13168 

99129 

;14896 

98884 

16620 

'98609 

26 

35 

09729 

99526  1146599341 

1319799125 

14925  98880 

16648 

98604 

25 

36 

09758 

99523  :  1149499337 

13226 

99122 

1495498876 

16677 

98600 

24 

37 

0978799520;  11523 

99334113254 

99118 

'14982  98871 

16706 

98595 

23 

38  09816  99517  11552 

99331113283 

99114 

15011198867 

16734 

,98590 

22 

39 

09845 

995141  11580 

99327 

13312 

99110I15040J98863 

16763 

'98585 

21 

40 

09874 

9951  ij  11609 

99324 

13341 

991061  15069'98858 

16792 

98580 

20 

41 

09903 

995081  11638 

99320 

13370 

99102115097198854 

16820 

98575 

19 

42 

09932 

995061  11667 

99317 

13399 

99098  15126  98849 

'16849 

98570 

18 

43 

09961 

'J95U3  11696 

99314 

13427199094  15155:98845 

1  16878  98565 

17 

44109990 

99500  11725  99310j|13456i99091  i  15184|98841 

16906  98561 

16 

45  10019 

99497  11754  99307||  13485 

99087J  15212  98836 

16935198556 

15 

46 

10048 

99494111783 

99303  13514  990S3i  15241  98832 

|16964|98551 

14 

47 

48 

10077 
10106 

99491 

99488 

;11812  99300  '13543  99079!  15270198827 
11840  992971'  13572199075!  15292  98823 

1699298546 
17021  98541 

13 
12 

49 
50 

10135199485 
10164199482 

11869  992931  13600  99071  15327:98818  17050  98536 
,11898  99290  13629  99067  15356!98814  17078  98531 

11 
10 

51 

10192  99479  11927 

99286  13658199063  15385  98809|!17107;98526 

9 

52 
53 

10221 
10250 

99476J  11956 
99473  11985 

99283  jl  3687 
99279  13716 

99059 
99055 

15414  98805  17136  90521 
15442198800|  17164  98516 

8 
7 

54 

10279199470  12014 

99276  :  13744 

99051 

15471 

98796 

17193:98511 

6 

56 

10308  99467 

12043 

99272 

113773 

99047  15500 

98791 

17222  98506 

5 

56 

1033799464  12071 

99269  13802 

99043115529 

98787 

1  7250  9850 

4 

57 

10366J  99461 

12100 

99265  13*31 

99039:15557 

98782 

17279  98496 

3 

58 

10395  99458 

12129 

99262':  13860 

99035(15586 

98778 

17308  9849 

2 

59 

104241  99455 

:12158 

99258;  1388S 

99031115615 

98773 

17336  '98486 

1 

M 

N.  CS.  N.  S. 

1  N.  CS. 

N.S.  X.cs. 

\.S.   N.CS. 

N.S. 

N.  CS 

N.S. 

M 

84  Deff. 

83  Dee.  11  «2  Deg.   |  81  Deg. 

80 

Ucg. 

04 


A   TABLE   OP   NATtHAL   SINES. 


10  Deg. 

I  1  1  Deg.. 

12  Deg. 

13  Deg. 

14  Deg. 

M 

N.S. 

N.  CS. 

I  N.S. 

N.  CS. 

N.S. 

N.  CS. 

N.S. 

N.  CS. 

N.S. 

N.CS. 

M 

'o 

17365 

98481 

;19081 

98163 

20701 

97815 

22495 

97437 

24192 

97030 

60 

1 

17393 

98476 

,19109 

98167 

20820 

97809 

22523 

97430 

24220 

97023 

59 

2 

17422 

98471 

19138 

98152 

20848 

97803 

22552 

97424 

24249  97015 

58 

3 

17451 

98466 

19167 

98146 

20877 

97797 

22580 

97417 

24277J97008 

57 

4 

17479 

98461 

:19195 

98140 

20905 

97791 

22608 

97411 

24305197001 

56 

5 

17508 

98465 

119224 

98135 

20933 

97784 

22637 

97404 

24333  96994 

55 

6 

17537 

98450 

19252 

98129 

20962 

97778 

22665 

97398 

24362  96987 

54 

7 

17565 

98445 

19281 

98124 

20990 

97772 

22693 

97391 

24390 

96930 

53 

8 

17594 

98440 

19309 

98118 

21019 

97766122722 

97384 

24418 

96973 

52 

9 

17623 

98435 

19338 

98112 

21047 

97760  j  22750 

97378 

24446 

96966 

51 

10 

17651 

98430 

19366 

98107 

21076 

97754 

22778 

97371 

24474  96959 

50 

11 

17680 

98425 

19395 

98101 

21104 

97748 

22807 

97365 

24503 

96952 

49 

12 

17708 

98420 

19423 

98096 

21132 

97742 

22835 

97358 

24531 

96945 

48 

13 

17737 

98414 

19452 

98090 

21161 

97735 

22863 

97351 

24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21189 

97729 

22892 

97345 

24587 

96930 

46 

15 

15794 

98404 

19509 

98079 

21218 

97723 

22920 

97338 

24615 

96923 

45 

16 

17823 

98399 

19538 

98073 

21246 

97717 

22948 

97331 

24644  '  969  16 

44 

17 

17852 

98S&4 

19566 

98067 

21275 

97711 

22977 

97325 

24672  96909 

43 

18 

17880 

98389 

19595 

98061 

21303 

97705 

23005 

97318 

24700  96902 

42 

19 

17909 

98383 

19623 

98056 

21331 

97698 

23033 

97311 

24728 

96894 

41 

20 

17937 

y8378 

19652 

98050 

21360 

97692 

23062 

97304 

24756 

96887 

40 

21 

17966 

98373 

19680 

98044 

21388 

97686 

23090 

97298 

24784 

96880 

39 

22 

17995 

98368 

19709 

98039 

21417 

97680 

23118 

97291 

24813 

96873 

38 

23 

18023 

98362 

19737 

98033 

21445 

97673 

23146  97284 

24841 

96866 

37 

24 

18052 

98357 

19766 

98027 

21474 

97667 

23175  97278 

24869 

96858 

36 

25 

18081 

98352 

19794 

98021 

21502 

97661 

23203)97271 

24897 

96851 

35 

26 

18109 

98347 

19823 

98016 

21530 

97655 

2323  1 

97264 

24925 

96844 

34 

27 

18138 

98341 

19851 

98010 

21559 

97648 

23260 

97257 

24953 

96837 

33 

28 

18166 

98336 

19880 

98004 

21587 

97642- 

23288 

97251 

24982 

96829 

32 

29 

18195 

98331 

19908 

97998 

21616 

97636 

23316 

97244 

25010 

96822 

31 

30 

18224 

98325 

19937 

97992 

21644 

97630 

23345 

97237 

25038 

96815 

30 

31 

18252 

98320 

19965 

979S7 

21672 

97623 

23373 

97230 

25066 

96807 

29 

32 

18281 

98315 

19994 

97981 

21701 

97617 

23401 

97223 

25094 

96800 

28 

33 

18309 

98310 

20022 

97975 

21729 

97611 

23429 

97217 

25122 

96793 

27 

34 

18338 

98304 

20051 

97969 

21758 

97604 

23458 

97210 

25151 

96786 

26 

35 

18367 

98299 

20079 

97963 

21786 

97598 

23486 

97203 

25179 

96778 

25 

iJti 

18395 

98294 

20108 

97958 

21814 

97592 

23514 

97196 

25207 

96771 

24 

37 

18424 

98288 

20136 

97952 

21843 

97585 

23542 

97189 

25235 

96764 

23 

38 

18452 

98283 

20165 

97946 

21871 

97579 

23571 

97182 

25263 

96756 

22 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

21 

40 

18509 

98272 

20222 

97934 

21928 

97566 

23627 

97169 

25320 

96742 

20 

41 

18538 

98267 

20250 

97928 

21956 

97560 

23656J971C2 

25348 

96734 

19 

42 

18567 

98261 

20279 

97922 

21985 

97553 

23684197155 

25376 

96727 

18 

43 

18595 

98256 

20307 

97916 

22013 

97547 

23712 

97148 

25404 

96719 

17 

44 

18624 

98250 

20336 

97910 

22041 

97541 

23740 

97141 

25432 

96712 

16 

45 

18652 

98245 

20-364 

97905 

22070 

97534 

23769 

97134 

25460 

96705 

15 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97127 

25488 

96697 

14 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

25516 

96690 

13 

48 

18738 

98229 

20450 

97887 

22155 

97515 

23853 

97113 

25545 

96682 

12 

49 

18767 

98223 

20478 

97881 

22183 

97508 

23882 

97106 

25573 

96675 

11 

50 

18795 

98218 

20507 

97875 

22212 

97502 

23910 

97100 

25601 

96667 

10 

51 

18824 

98212 

20535 

97869 

22240 

97496 

23938 

97093 

25629 

96660 

9 

52 

18852 

98207 

20563 

97863 

22268 

97489 

23966 

97086 

25657 

96653 

8 

53 

18881 

98201 

20592 

97857 

22297 

97483 

23995 

97079 

25685196645 

7 

54 

18910 

98196 

20620 

97851 

22325 

97476 

24023 

97072 

25713 

96638 

6 

55 

18938 

98190 

20649 

97845 

22353 

97470 

24051 

97065 

25741 

96630 

5 

56 

18967 

98185 

20677 

97839 

22382 

97463 

24079 

97058 

25769 

96623 

4 

57 

18995 

98179 

20706; 

97833 

22410 

97457 

24108 

97051 

25798 

96615 

3 

58 

19024 

98174 

20734 

97827 

22438 

97450 

24136 

97044 

25826 

96608 

2 

59 

19052 

98168 

20763 

97821 

22467 

97444 

24164 

97037 

25854 

9660P 

1 

M 

N.  CS.  |  N.  S. 

N.CS. 

N.S. 

N.  CM. 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

M 

79  Deg. 

78  Deg. 

77  Deg. 

76  Deg.  | 

75  Deg. 

A   TABLE    OF   NATURAL    SINES. 


15  Dej/. 

16  Deg.  , 

J7  i>eg. 

18  Deg. 

19  Deg. 

M 

X.  6. 

N.  CS. 

N.  3. 

N.  CS. 

X.  S. 

i\.  CS. 

N.S. 

N.CS. 

N.S. 

N  CS 

H 

~0 

25882 

96593 

27564 

96126 

29237 

95630 

30902 

95106 

32557 

94552  60 

1 

25910 

96585 

27592 

96118 

29265 

95622 

30929 

95097 

325S4 

94542  59 

2 

26938 

96578 

27620 

96110' 

29293 

95613 

30957 

95088  32612 

9453368 

3 

25966 

96570 

27648 

96102 

2932  1  95605 

30985 

95079J32639 

94523  57 

4 

25994 

96562 

27676 

96094 

29348195596 

31012 

95070..32667 

9451456 

5 

26022,96555 

27704 

96086 

29376  95588 

31040 

9  506  ill  32694 

94504  55 

6 

26050196547 

27731 

96078 

29404(95579 

31068 

95052!  i  32722 

94495 

54 

7 

26079  96540 

27759 

96070 

29432,95571 

31095 

96043132749 

94485 

53 

8 

26107  96532 

27787 

96062 

2946095562 

31123 

95033' 

32777 

94476152 

9 

26135 

96524 

27*15 

96054 

29487 

95554 

31151 

95024 

32804 

94466  51 

10 

26163 

96517 

27843 

96046 

29515 

95545 

31178 

95015 

32832 

94457 

50 

il  :>6191  96509 

2787-1 

96037 

29543 

95536 

31206 

95006 

32859 

94447 

49 

1226219  96502 

27899 

96029 

29571 

95528 

31233 

94997 

32887 

94433 

48 

13  26247  96494 

27927 

96021 

29599  95519 

31261 

94988  32914 

94428 

47 

14 

26275  96486 

27955 

96013 

2962695511 

31289 

94979  32942 

94418 

46 

15)26303  96479 

27983 

96005 

29654  95502 

31316 

94970 

32969 

94409 

45 

16 

26331  9647] 

280  1  1 

95997 

29682 

95493 

3134494961 

32997 

94399 

44 

17 

26359  96463 

28039 

95989 

29710 

95485 

31372194952 

33024 

94390 

43 

18  26337  96456 

28067 

95981 

29737 

95476 

31399 

94943 

33051 

94380 

42 

19 

26415  96448  1  28095 

95972 

29765  95467 

31427 

94933 

33079 

94370 

41 

20 

26443  96440  28123 

95964 

29793j95459 

31454 

94924 

33106 

94361 

40 

21 

2647  lj  96433 

28150 

95956 

29821 

95450 

31482 

94915 

33134 

94351 

39 

22 

2650096425 

28178 

95948 

29849 

95441 

31510 

94906 

33161 

94342 

38 

23 

26528  96417 

28206 

95940 

29876  95433 

31537 

94897 

33189 

94332 

37 

24 

2655696410 

28234 

95931 

29904  95424 

31565 

94888 

33216 

94322 

36 

25  26584196402 

28262 

95923 

29932 

95415 

31593 

94878 

33244 

94313 

35 

262661296394 

28290 

95915 

29960 

95407 

31620 

94869 

33271 

94303 

34 

2726640 

96386 

28318 

95907 

29987 

95398 

31648 

94860 

33298 

94293 

33 

28  26668 

96379 

28346 

95898 

30015 

95389 

31675 

94851 

33326 

94284 

32 

29  26696  96371 

28374 

95890 

30043 

953SO 

31703 

94842 

33353 

94274 

31 

30 

26724  96363 

28402 

95882 

30071 

95372 

31730 

94832 

33381 

94264 

30 

31 

26752  96355 

28429 

95874 

30098 

95363 

31758 

94823 

3340S 

94254 

29 

32  26780 

96347 

28457 

95865:30126 

95354 

31786 

94814 

33436 

94245 

28 

33:26808 

96340 

28485 

95857 

30154 

95345 

31813 

94805 

33463 

94235 

27 

34  26836 

96332 

28513 

95849  30182 

95337 

31841 

94795 

33490 

94225 

26 

35 

26864 

96324 

28541 

95841  30209 

9532* 

31868 

94786 

33518J94215 

25 

36 

26892 

96316 

28569 

95832130237 

95319 

31896 

94777 

33545  94206 

24 

37 

26920  96308 

28597 

95824JI30265 

95310 

31923 

94768 

33573 

94196 

23 

38 

26948)96301 

28625 

95816 

30292 

95301 

31951 

94758 

33600 

94186 

22 

39  26976  96293 

28652 

95807 

30320 

95293 

31979 

94749 

33627 

94176 

21 

40  27004  96285 
41  2703296277 

28680 
28708 

95799  30348 
95791  30376 

95284 
95275 

32006 
32034 

94740 
94730 

33655 
33682 

94167 
94157 

20 
19 

42 

27060:96269 

28736 

95782  30403 

95266 

32061 

94721 

33710 

94147 

18 

43i27088i96261 

28764 

95774 

30431 

95257 

32089 

94712 

33737 

94137 

17 

44 

27116  96253 

28792 

95766  30459 

95248 

32116 

94702 

33764 

94127 

16 

45  27144  96246 

28820 

95757)30486 

95240 

32144 

94693 

33792 

94118 

15 

4627172 

96238 

28847 

95749  30514 

95231 

32171 

94684 

33819 

94108 

14 

47 

27200 

96230 

28875 

95740:  30542 

95222 

32199 

94674 

33846  194098 

13 

48 

27228)96222 

28903 

95732i;30570 

95213 

32227 

94665 

33874  J94088 

12 

49127256i96214 

28931  95724ii30597 

95204 

32254 

94656 

33901  194078 

11 

50,  27284  1  96206 

28959  95715  :  30625 

95195 

32282 

94646 

13392994068 

10 

51 

27312  96198 

28987  95707  \\  30653 

95186 

32309 

946371.3395694058 

9 

52 

27340 

96190 

29015 

95698  30680 

95177 

32337 

94627  33983  94049 

8 

53 

27368 

96182 

29042 

95690;  30708 

95168 

32364  94618 

34011  94039 

7 

5427396 
5527424 

96174 
96166 

29070 
29098 

95681  |!30736|95159ii32392  94609 
95673  !30763!95150l  32419  194599 

34038  94029 
34065  94019 

6 
5 

56/^7452 

57.27480 

96158 
96150 

29126}95664|  30791  95142IJ32447 
29154  95656i'30819  95133  32474 

94590 
94580 

34093  94009 
34120  93999 

4 
3 

58  27508 

96142 

29182  95647i|30846  95124  132502 

94571 

34147  93989 

2 

59i2753fi 

96134 

29209  95639!  30874 

96116  32529 

94561 

34175 

93979 

I 

.M 

N.  CS.  N.  S. 

N.  CS.  1  N.  S.  j  N.  C8. 

N.  S.  IN.CS. 

N.S. 

N.  CS. 

sfis 

74  Deg. 

73  Deg.  1  72  Deg.   1  71  Dei?,  il  70  De?.  | 

96 


A   TABLE    OF   NATUBAL   SINES. 


20  Ueg. 

21  Deg. 

22  Deg. 

23  Deg. 

24  Deg. 

M 

N.  S. 

N.  CS. 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

N.CS. 

M 

0 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92050 

40674 

91355 

60 

1 

34229 

93959135864 

93348 

37488 

92707 

139100 

92039 

40700 

91343 

59 

2 

34257 

93949  35891 

93337 

37515 

92697 

39127 

92028 

40727 

91331 

58 

3 

34284 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91319 

57 

4 

34311 

93929 

35945 

93316 

37569 

92675 

J39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

!39207 

91994 

40806 

91295 

55 

6 

34366 

93909 

36000 

93295 

37622 

92653 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

53 

8 

34421 

93889 

36054 

93274 

37676 

92631 

39287 

91959 

40886 

91260 

52 

9 

34448 

93879 

36081 

93264 

37703 

92620 

39314 

91948 

40913 

91248 

51 

10 

34475  j  93869 

36108 

93253 

37730  92609 

39341 

91936 

40939 

91236 

50 

11 

34503 

93859 

36135 

93243 

37757 

92598 

39367 

91925 

40966 

91224 

49 

12 

34530 

93849 

36162 

93232 

37784 

92587 

39394 

91914 

40992 

91212 

48 

13 

34557 

93839 

36190 

93222 

37811 

92576 

39421 

91902 

41019 

91200 

47 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91891 

41045 

91188 

46 

15 

34612 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

16 

34639 

93809 

36271 

93190 

37892 

92543 

39501 

91868 

4~1098 

91164 

44 

17 

34666193799 

36298 

93180 

37919 

92532 

39528 

91856 

41125 

91152 

43 

18 

34694193789 

36325 

93169 

37946 

92521 

39555 

91845 

41151 

91140 

42 

19 

34721 

93779 

36352 

93159 

37973 

92510 

39581 

91833 

41178 

91128 

41 

20 

34748 

93768 

36379 

93148 

37999 

92499  39608 

91822 

41204 

91116 

40 

21 

34775 

93759 

36406 

93137 

38026 

92488 

39635 

91810 

41231 

91104 

39 

22 

34803 

93748 

36434 

93127 

38053 

92477 

39661 

91799 

41257 

91092 

38 

23 

34830 

93738 

36461 

93116 

38080 

92466 

39688 

91787 

41284 

91080 

37 

24 

34857 

93728 

36488 

93106 

38107 

92455 

39715 

91775 

41310 

91068 

36 

25 

34884 

93718 

36515 

93095 

38134 

92444 

39741 

91764' 

41337 

91056 

35 

26 

34912 

93708 

36542 

93084 

38161 

92432 

39768 

91752 

41363 

91044 

34 

27 

34939 

93698 

36569 

93074 

38188 

92421 

39795 

91741 

41390 

91032 

33 

28 

34966 

93688 

36596 

93063 

38215 

92410 

39822 

91729 

41416 

91020 

32 

29 

34993 

93677 

36623 

93052 

38241 

92399 

39848 

91718 

41443 

91008 

31 

30 

35021 

93667 

36650 

93042 

38268 

92388 

39875 

91706 

41469 

90996 

30 

31 

35048 

93657  36677 

93031 

38295 

92377 

39902 

91694 

41496 

90984 

29 

32 

35075 

93647  36704 

93020 

38322 

92366 

39928 

91683 

41522 

90972 

28 

33 

35102 

93637  36731 

93010 

38349 

92355 

39955  91671 

41549 

90960 

27 

34 
35 

35130 
35157 

93626  36758 
93616  36785 

92999 
92988 

38376 
38403 

92343 
92332 

39982  91660 
40008  91648 

41575 
41602 

90948 
90936 

26 
25 

36 

35183 

93606 

36812 

92978 

38430 

92321 

40035 

91636 

41628 

90924 

24 

37 

35211 

93596 

36839 

92967 

38456 

92310 

40062 

91625 

41655 

90911 

23 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

91613 

41681 

90899 

22 

39 

35266 

93575  36894 

92945 

38510 

92287 

40115 

91601 

41707 

90887 

21 

40 

35293 

93565  36921 

92935 

38537 

92276 

40141 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

91578 

41760 

90863 

19 

42 

35347 

93544 

36975 

92913 

38591 

92254 

40195 

91566 

41787 

90851 

18 

43 

35375 

93534 

37002 

92902 

38617 

92243 

40221 

91555 

41813 

90839 

17 

44 

35402 

93524 

37029 

92892 

38644 

92231 

40248 

91543 

41840 

90826 

16 

45 

35429 

93514 

37056 

92881 

38671 

92220 

40275 

91531 

41866 

90814 

15 

46 

35456 

93503 

37083 

92870 

38698 

92209 

40301 

91519 

41892 

90802 

14 

47 

35484 

93493 

37110 

92859 

38725 

92198 

40328 

91508 

41919 

90790 

13 

48 

35511 

93483 

37137 

92849 

38752 

92186 

40355 

91496 

41945 

90778 

12 

49 

35538 

93472 

37164 

92838 

38778 

92175 

40381 

91484 

41972 

90766 

11 

50 

35565 

93462 

37191 

92827 

38805 

92164 

40408 

91472 

41998 

90753 

10 

51 

35592 

93452 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

52 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42051 

90729 

8 

53 

35647 

93431 

37272 

92794 

38886 

92130 

40488 

91437 

42077 

90717 

7 

54 

35674 

93420 

37299 

92784 

38912 

92119 

40514 

91425 

42104 

90704 

6 

55 

35701 

93410 

37326 

92773 

38939 

92107 

40541 

91414 

42130 

90692 

5 

56 

35728 

93400 

37353 

92762 

38966 

92096 

40567 

91402 

42156  J90680 

4 

57 

35755 

93389 

37380 

92751 

38993 

92085 

40594 

91390 

42183  90668 

3 

58 

35782 

93379 

37407 

92740 

39020 

92073 

40621 

91378 

42209  90655 

2 

59 

35810 

93368 

37434 

92729 

39046 

92062 

40647 

91366 

42235 

90643 

1 

Mf 

N.  OS.  |  N.  S. 

N.CS. 

N.S". 

N.  CS7 

N.S. 

NTcs: 

N.S. 

N.  CS. 

N.S. 

M 

69  Deg. 

68  Deg. 

67  Deg. 

66  Deg. 

65  Deg, 

A    TABLE    OP    NATURAL    SINES. 


25  Deg. 

26  Deg.  | 

27  L»eg. 

2ii  Deg. 

29  Deg. 

M 

N.S. 

N.  <JS. 

IsCsT 

N.  CS. 

N.S. 

N.  CS. 

N.S.  |N.CS. 

N.S.  |N.CS. 

M 

0 

42262 

90631 

43837 

89879 

45399 

89101 

46947  88295 

48481 

87462  GO 

1 
2 
3 

42288 
42315 
42341 

90618 
90606 

90594 

43863 
438S9 
43916 

89867 
89854 
89841 

45425 
45451 
45477 

89087 
89074 
89061 

46973  8823  1 
46999188267 
47024  88254, 

48506  87448  59 
48532  87434  58 
48557  87420  57 

4 

42367 

905S2 

43942 

89828 

45503 

89048 

47050 

88240j  48583 

87406  56 

5 

42394 

90569 

43968 

89816 

45529 

89035 

47076 

88226  48608 

8739  Ii55 

6 

42420 

90557 

43994 

89803 

45554 

89021 

47101 

88213,48634 

87377  54 

7 

42446 

90545 

44020 

89790 

45580 

89008 

47127 

88  199  [48659 

8736353 

8 

42473 

90532 

44046 

89777 

45606 

88995 

47153 

8S185 

48684 

87349,53 

9 

42499 

90520 

14072 

89764 

45632 

88981- 

47178 

88172 

48710 

87335151 

10 

42525 

90507 

44098 

89752 

45658 

88968 

47204 

88158  |48735 

87321  50 

11 

42552 

90495 

44124 

89739 

45684 

88955 

47229 

88144 

48761 

87306  49 

12 

42578 

90483 

44151 

89726 

45710 

88942 

47255 

88130 

48786 

87292  48 

13 

42604 

90470 

44177 

89713 

45736 

88928 

47281 

88117 

48811 

87278147 

14 

42631 

90458 

44203 

89700 

45762 

88915 

47306 

88103| 

48837 

87264146 

15 

42657 

90446 

44229 

89687 

45787 

88902 

47332 

88089 

48862 

87250,45 

If.  42683 

90433 

44255 

89674 

45813 

88888 

47358 

83075 

48888 

87235  44 

17 

42709 

90421 

44281 

89662 

45839 

88875 

47383 

88062 

48913 

87221 

43 

18 

42736 

90408 

44307 

89649 

45865 

88862 

47409 

88048 

48938 

87207J42 

19 

42762 

90396 

44333 

S9636 

45891 

88848 

47434 

88034 

48964 

8719341 

20 

42788  90383 

44359 

39623 

45917 

88835 

47460 

88020J 

48989 

87178  40 

21 

4281590371 

44385 

39610 

45942 

88822 

47486 

88006| 

49014 

8716439 

22 

42S41 

90358 

44411 

89597 

45968 

88808 

47511 

87993! 

49040 

87150  38 

23 

42867 

90346 

44437 

S9584 

45994 

88795 

47537 

87979! 

49065 

8713637 

24J42894 

90334 

44464 

89571 

46020 

88782 

47562 

87965 

49090 

87121  36 

25 

42920190321 

44490 

39558 

46046  88768 

47588 

87951 

49116 

87107 

35 

26 

42946 

90309 

44516 

39545 

46072 

88755 

47614 

87937 

49141 

87093 

34 

27 

42972 

90296 

44542 

89532 

46097 

88741 

47639 

87923 

49166 

87079 

33 

28 

42999 

90284 

44568 

89519 

46123 

88728 

47665 

87909 

49192 

87064 

32 

29 

43025  90271 

44594 

895v>6 

46149 

88715 

47690 

87896 

49217 

87050 

31 

30 

43051 

90259 

44620 

89493 

46175 

88701 

47716 

87882 

49242 

87036 

30 

31 

430?7 

90246 

44646 

39480 

46201 

88688 

47741 

87868 

49268 

87021 

29 

32 

43104 

90233 

44672 

39467 

46226 

88674 

47767 

87854 

49293 

87007 

28 

33 

43130 

90221 

44698 

39454 

46252 

88661 

47793 

87840 

49318 

86993 

27 

34 

43156 

90208 

44724 

89441 

46278 

88647 

478  18 

87826 

49344 

86978 

26 

3o 

431GQ  OfilQfi 

A  4.  760 

(394*8 

40304 

88O34 

47844 

8/812 

49369 

86964 

25 

36 

43209  90183 

44776 

89415 

46330 

88620 

47869 

87798 

49394 

86949 

24 

374323590171 

44802 

89402  J46355 

88607 

47895 

87784 

49419 

86935 

23 

38i4326l!90158 

44828 

89389  1!46381 

88593 

47920 

87770 

49445 

86921 

22 

39  43287 

90146 

44854 

89376 

46407 

88580 

47946 

87756 

49470 

86906 

21 

4043313 

90433 

44880 

89363 

46433 

88566 

47971 

87743 

49495 

86892 

20 

4  1  143340 

90120 

44906 

893oO  ,46458 

88553 

47997 

87729 

49521 

86878 

19 

42  43366 

901G8 

44932 

89337146484 

88539 

48022 

87715 

49546 

86863 

18 

43143292 

90095 

44958 

89324 

46510 

88526 

48048 

87701 

49571 

86849 

17 

44'43418i90082 

44984 

89311 

46536 

88512 

48073 

87687 

49596 

86834 

16 

45 

43445  j  90070 

45010 

89298 

46561 

88499 

48099 

87673 

49622 

86820 

15 

46 

43471 

90057 

45036 

89285 

46587 

88485 

48124 

87659 

49647 

86805 

14 

47 

43497 

90045 

45062 

89272  46613 

88472 

48150 

87645 

49672 

86791 

13 

48 

43523 

90032 

45088 

89259  46639 

88458 

48175 

87631 

49697J86777 

12 

40 

43549 

90019 

45114 

89245,  46664 

88445 

4820] 

8761? 

49723186762 

11 

50 

43575 

90007 

45140  89232  146690 

88431 

48226 

87603 

49748 

86748 

10 

51143602 

89994 

45166 

89219  46716 

88417 

48252 

87589 

49773 

86733 

9 

52143628 

89981 

45192 

89206  46742 

88404 

48277  187575 

49798 

86719 

8 

53  43654 

89968 

45218 

89193  46767 

88390 

48303  '87561 

49824 

86704 

7 

54|43680 

89956 

45243 

89180  46793 

88377 

48328187546 

4984986690 

6 

55143706 

89943 

45269 

89l67i468l9J88363 

48354187532 

49874  86675 

5 

56|43733 

89930 

45295 

89153  4684488349 

48379 

37518 

49899  86661 

4 

57J43759 

89918 

45321 

*<J140  4687088336 

48405 

37504 

49924  86646 

3 

58 

43785 

89905 

45347 

89127 

46896  88322 

48430 

87490 

49950 

86632 

2 

5S 

43811 

89892 

45373 

89114 

46921  88308 

48456 

87476 

49975 

86617 

1 

M 

N.CS. 

N.S. 

N^ST 

N.S.  ||  N.CS.  I  N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

M 

64  Deg. 

63  Deg.  1J  62  Deg. 

61  Deg. 

60  Deg. 

98 


A   TABLE    OF    NATURAL    SINES. 


30  JDeg. 

|  31  Deg. 

32  Ueg.  ||  33  Deg.  | 

34  Deg. 

M 

N.  S. 

N.  CS. 

!  N.S. 

N.  CS. 

N.S.  N.  CS.  j  N.S. 

N.  CS.  : 

N.S. 

N.  CS. 

M 

0 

50000 

86603 

51504 

85717 

52992  84805;  54464 

83867 

55919 

82904 

60 

1 

50025186588 

51529 

85702 

53017 

84789  J54488 

83851 

55943 

82887 

59 

2 

50050 

86573 

51554 

85687 

53041 

84774  !  545  13 

83835  I55968J82871 

58 

3 

50076 

86559  51579 

85672 

5306684759(54537 

83819155992  82855 

57 

4 

50101 

86544  51604 

85657 

53091  84743  1  54561 

83804)5601682839 

56 

5 

50126 

86530 

51628 

85642 

5311584728)54586 

83788> 

56040  82822 

55 

6 

7 

50151(86515 

50176186501 

51653 

51678 

85627 
85612 

5314084712 
5316484697 

54610 
54635 

83772 
83756 

56064 
56088 

82306 
82790 

54 
53 

8 

50201 

86486 

51703 

85597 

53189 

84681 

54659 

83740 

56112 

82773 

52 

9 

50227 

86471 

51728 

85582 

53214 

84666 

54683 

83724 

56138 

82757 

51 

10 

50252)86457 

51753 

85567 

53238  84650 

54708 

83708 

56160 

82741 

50 

11 

5027786442 

51778 

8555  1 

53263)84635 

54732 

83692 

56184 

82724 

49 

12 

50302)86427 

51803 

85536 

53288  84619 

54756 

83676 

56208 

8270S 

48 

13 

50327 

86413 

51828 

85521 

53312184604 

54781 

83660 

56232 

82692 

47 

14 

50352 

86398 

51852 

85506 

53337  84588 

54805 

33645 

56256 

82675 

46 

15 

50377 

86384 

51877 

85491 

53361)84573 

54829 

83629 

56280 

82659 

45 

16 

50403 

86369 

51902 

85476 

53386 

84557 

54854 

83613 

56305 

82643 

44 

17 

50428 

86354 

51927 

85461 

53411 

84-542 

54878 

83597 

56329  82626 

43 

18 

50453 

86340 

51952 

85446 

53435,84526 

54902 

83581 

56353182610 

42 

19 

50478 

86325 

51977 

85431 

5346018451  1 

54927 

83565 

56377J82593 

41 

20 

50503 

86310 

52002 

85416 

53484)84495 

54951 

83549 

5640182577 

40 

21 

50528  i  86295 

5202G 

85401 

53509 

84480 

54975 

83533 

56425  '82561 

39 

22 

5055386281 

52051 

85385 

53534 

84464 

54999 

83517 

56449 

82544 

38 

23 

50578 

86266 

52076 

85370 

53558 

84448 

55024 

83501 

56473 

82528 

37 

24 

50603 

86251 

52101 

85355 

53583 

84433 

55048 

83485 

56497 

82511 

36 

25 
26 

50628 
50654 

86237 
86222 

52126 
52151 

85340 
85325 

53(M)7 
53632 

84417 
84402 

55072 
55097 

83469 
83453 

56521  82495 
56545  82478 

35 
34 

27 

50679 

86207 

52175 

85310 

53656 

84386 

55121 

83437 

56569 

82462 

33 

28 

50704 

86192 

52200 

85294 

53631 

84370 

55145 

83421 

56593 

82446 

32 

29 

50729 

86178 

52225 

85279 

53705 

S4355 

5M69 

83405 

56617 

82429 

31 

30 

50754 

86163 

52250 

85264 

53730 

84339 

55194 

83389 

56641 

82413 

30 

31 

50779 

86148 

52275 

85249 

53754 

84324 

55218 

83373 

56665 

82396 

29 

32 

50804 

86133 

52299 

85234 

53779 

84308 

55242 

83356 

56689 

82380 

28 

33 

50829 

86119 

52324 

85218 

53804 

84292 

55266 

83340 

56713 

82363 

27 

34 

50854 

86104 

52349 

85203 

53828 

84277 

55291 

83324 

56736 

82347 

23 

35 

50879 

86089 

52374 

aoiss 

oasc>3 

S4Z01 

OO310 

833O8 

567«O 

62330 

25 

36 

50904 

86074 

52399 

85173 

53877 

84245 

55339 

83292 

56784 

82314 

24 

37 

50929 

86059 

52423 

85157 

53902 

84230 

55363 

83276 

56808 

82297 

23 

38 

50954 

86045 

52448 

85142 

53926 

84214 

55388 

83260 

568312)82281 

22 

39 

50979 

86030 

52473 

R5127 

53951 

84198 

55412 

83244 

56856)82264 

21 

40 

51004 

86015 

52498 

85112 

53975 

84182 

55436 

83228 

56880)82248 

20 

41 

51029 

86000 

52522 

85096 

54000 

84167 

55460 

83212 

56904 

82231 

19 

42 

51054 

85985 

52547 

85081 

54024 

84151 

55484 

83195 

56928 

82214 

18 

43 

51079 

85970 

52572 

85066 

54049 

84135 

55509 

83179 

56952 

82198 

17 

44 

51104 

85956 

52597 

85051 

54073 

84120 

55533 

83163 

56976 

82181 

16 

45 

51129 

85941 

52621 

85035 

54097 

84104 

55557 

83147 

5700082165 

15 

46 

51154 

85926  52646 

85020 

54122 

84088 

5558  1 

83131 

57024182148 

14 

47 

51179 

859111152671 

85005 

54  1  40 

84072 

55605 

83115 

5704788132 

13 

48 
49 

51204 
51229 

85896)52696 
85881  52720 

84989 
84974 

54171 
54195 

84057 
84041 

55630 
55654 

83098 
83082 

5707  1)82115 
57095^82098 

12 
11 

50 

51254 

85866 

52745 

84959 

54220 

84025 

55678 

83066 

57119*82082 

10 

51 

51279 

85851 

52770 

84943 

54244 

84009 

55702 

83050 

5714382065 

9 

52 

51304 

85836 

52794 

84928  1:5426983994 

55726 

83034 

571671820-8 

8 

53 

51329 

85821 

52819 

84913 

i54293|83978 

55750 

83017 

57191  82032 

7 

54 

51354 

85806  152844 

84897 

i54317 

83962 

55775 

83001 

57215^82015 

6 

55 

51379 

85792 

52869 

84882 

i54342 

83946 

56799 

82985 

57238  81999 

5 

56 

51404 

85777 

52893 

84866  54366 

83930 

55823 

82969 

57262  81982 

4 

57 

51429 

85762 

52918 

84851 

154391(83915 

55847182953 

57286181965 

3 

58 

51454 

85747 

52943 

84836 

15441583399  5587i}82936 

573  10  i8  1949 

2 

59 

51479 

85732 

52967 

84S20i;  54440 

83883  j  55895 

82920 

57334 

81932 

1 

M 

N.  CS. 

N.  S. 

N.  CS. 

N.  S.   N.  OS. 

N.S.   N.  CS. 

N.S. 

IN.CS. 

N.S. 

M 

59  Deg. 

58  Deg.   ii  57  Dc<r.   II  56  Dejj. 

55  Dee- 

A   TABLE    OF    NATURAL    SINES. 


35  Deg. 

36  De#. 

37  L>eg. 

'M  Deg. 

39  Deg. 

M 

N.S. 

N.  CS. 

N.S.  jN.O. 

N.S.  IN.CS. 

N.S. 

N.  CS. 

N.S. 

N.  CS. 

M 

0 

57358 

81915 

58779[80902 

60182 

79864 

61566 

7~880l 

62932 

77715 

60 

1 

57381 

81899 

58802(80885 

60205 

79846 

61589 

78783 

62955  !  77696 

59 

2 

57405 

81882 

58826 

80867 

60228 

79829 

61612 

78765 

62977177678 

58 

3  57429 

4  57453 

81865 
81848 

58849 
58873 

80850 
80833 

60251 
60274 

7G811 
79793 

61635 
61658 

78747 
78729 

63000 
63022 

77660 
77641 

57 
56 

5 

57477 

81832 

58896 

80816 

60298 

79776 

61681 

78711 

63045 

77623 

55 

6 

57501 

81815 

58920 

80799 

60321 

79758 

61704 

78694 

63068 

77605 

54 

7 

57524 

81798 

58943 

80782 

60344 

79741 

61726 

786761:63090 

77586 

53 

8 

57548 

81782 

58967 

807C5 

60367 

79723 

61749 

78668  63113 

77568 

52 

9 

57572 

81765 

58990 

80748 

60390 

79706 

61772 

78640  63135 

77550 

51 

10 

57596 

81748 

59014 

80730 

60414 

79688 

61795 

78622  63158 

77531 

50 

11 

57619 

81731 

59037 

80713 

60137 

79671 

61818 

78604  63180 

77513 

49 

12  57643 

81714 

59061 

80C96 

60460 

79G53 

61841 

78536 

63203 

77494 

48 

13157667 

81698 

59084 

80679 

69483 

79635 

61864 

78568 

63225 

77475 

47 

1457691 

81681 

59108 

80662 

60506 

79618 

61887 

78550 

63248 

77458 

46 

15157715 

81664 

59131 

80S44 

60529 

79600 

61909 

78532 

63271 

77439 

45 

1657738 

81647 

59154 

30627 

60553 

79583 

61932 

78514 

63293 

77421 

44 

17|57762 

81631 

59178 

80610 

6057G 

79565 

61955 

78496 

63316 

77402 

43 

18J57786 

81614 

59201 

80593 

60599 

79547 

61978  78478 

63338 

77384 

42 

19  57810 

81597 

59225 

80576 

60622 

79530 

62001  78460 

63361 

77366 

41 

20  57833 

81580 

59248 

80558 

60645 

79512 

62024 

78442 

63383 

77347 

40 

21  57857 

81563 

59272 

80541 

60668 

79494 

62046 

78424 

63406 

77329 

39 

22  57*i<l 

81546 

59295 

80524 

60691 

79477 

62069 

78405 

63428 

77310 

38 

23  57904 

81530 

59318 

80507 

60714 

79459 

62092 

78387 

63451 

77292 

37 

24 

57928 

81513 

59342 

80489i 

60738 

79441 

62115 

78369 

63473 

77273 

36 

25 

57952 

81496 

59365 

80472! 

60761 

79424 

62138 

78351 

6«>496 

77255 

35 

26  57976 

81479 

59389  80455 

60784 

79406 

62160 

73333 

63518 

77236 

34 

27  57999|81462 

59412  80438 

60807 

79338 

62183 

78315 

63540 

77218 

33 

28  5802381445  59436 

80420j 

60830 

79371 

62206 

78297 

63563 

77199 

32 

29*58047  814281159459 

80403i 

60853 

79353 

62229 

78279 

63585 

77181 

31 

30158070 

81412 

|59482 

80386. 

60876 

79235 

62251 

78261 

63608 

77162 

30 

31  58094 

81395 

59506 

80368 

60399 

79318 

62274 

78243 

63630 

77144 

29 

3258118 

81378 

59529 

80351 

60922 

79300 

62297 

78225 

63653 

77125 

28 

3358141 

81361 

59552 

80334 

60945 

79282 

62320 

782061 

63675 

77107 

27 

34158165 

81344 

159576 

80316 

60968 

79264 

62342 

78188 

63698 

77088 

26 

35(58189 

81327 

59599 

80299 

60991 

79247 

62365 

78170 

63720 

77070 

25 

36  58212 

81310 

59622 

80282 

61015 

79229 

62388 

78152 

63742 

77051 

24 

3758236 

81293 

59646 

80264 

61038 

79211 

82411 

78134 

33765 

77033 

23 

38|58260 

81276 

59669 

80247  61061 

79193 

62433 

78116 

63787 

77014 

22 

39  58283 

81259 

159693 

80230  j  61084 

79176 

62456 

78098 

63810 

76996 

21 

40;58307 

81242 

159716 

80212  61107 

79158 

62479 

78079 

63832 

76977 

20 

41  58330 

81225 

59739 

S0195  61130 

79140 

62502 

78061 

63854 

76959 

19 

42  .->S3.->1. 

81208 

59763 

80178  61153 

79122 

62524 

78043 

63877176940 

18 

43  58378 

81191 

59786 

S0160  61176 

79105 

62547 

78025 

63899 

76921 

17 

44  58401 

81174 

59809 

80143  61199 

79087 

62570 

78007 

63922 

76903 

16 

45 

58425 

81157 

59832 

80125  61222 

79069 

62592 

77988 

63944  76884 

15 

46  58449 

81140j5!H58 

80108  61245 

79051 

62615 

77970 

63966176866 

14 

47158472 

81123  59879 

80091 

61268 

79033 

62638 

77952 

63989|76847 

13 

48  .53496 
49  58519 

81106 
81089 

59902 
59926 

80073  61291 
80056  61314 

79015 
78998 

62660 
62683 

77934 
77916 

64011  76823 
64033  76810 

12 
11 

50  58543 

81072 
81055 

59949  80038  J61337 
59972  8002  1,|  6  1360 

78980 
78962 

62706 

62728 

77897 
77879 

6405676791 
64078176772 

10 
9 

52  5S590 
53  5^614 

81038 
81021 

5999580003  01  383 
60019  799861J61406 

78944 
78926 

62751 
62774 

77861 
77843 

6410076754 
64123  76735 

8 
7 

81004 

60042  79968  61429 

78908 

62796  '77824 

64145  76717 

6 

55|58661 

80987 

60065  j  79951  16  1451 

78891 

62819J77806 

64167  76698 

5 

56)58684 

80970 

60089  79934  'G  1474 

7S873 

62842  77788 

164190  76679 

4 

57 

58708 

80953 

60112  79916  ,61497  7SS55 

B28G4  77769(642  12.76661  3 

58 

58731 

80936 

60135  79S99  61520 

73837 

62887  77751  164234  76642 

2 

59 

58755 

80919 

60158  79881161543  78819 

62909  77733  1|64256  76623 

1 

M 

N.  CS.  N.  S. 

N.  CS.  I  N.  S.  J  N.  CS.  I  N.  S. 

N.  CSf 

N.8.  RN.ca 

N.S. 

M 

54  Deg. 

53  Deg.  1  52  Deg. 

51  Deg.  (I  50  De*. 

100 


A   TABLE   OF   NATUKAL   SINES. 


40  IVf  . 

41  Deg. 

42  Ueg. 

i  43  Deg. 

44  Deg. 

M 

N.  S. 

N.  CTJ 

N.S: 

N.  CS. 

N.S. 

N.  CS. 

N.S. 

N.  CS. 

N.S. 

N.  CS. 

M 

0 

64279 

76604 

65606 

75471 

66913 

74314 

68200 

73135 

69466 

71934 

60 

1 

64-301 

76586 

65628 

75452 

66935 

74295 

68221 

73116 

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TTcs.  !  N.  S. 

N.  CS. 

N.S. 

N.CS.I  N.S. 

NTcs: 

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N.S 

Mf 

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48  Deg. 

47  Deg. 

46  Deg. 

45  Deg.  j 

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